germanium and gold wire arrays embedded in silica photonic ... · germanium and gold wire arrays...

Germanium and gold wire arraysembedded in silica photonic crystal

fibers

Der Naturwissenschaftlichen Fakultät

der Friedrich-Alexander-Universität Erlangen-Nürnberg

zur

Erlangung des Doktorgrades

Vorgelegt von

Hemant Kumar Tyagi

aus Meerut, India

Als Dissertation genehmigt von der NaturwissenschaftlichenFakultät der Friedrich-Alexander Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 24 January 2011

Vorsitzenderder Promotionskommission: Prof. Dr. Rainer Fink

Erstberichterstatter: Prof. Dr. Philip St.J. Russell

Zweitberichterstatter: Prof. Dr. Tim Birks

ii

Contents

1. Introduction 7

2. Basic theory for waveguiding 11

2.1. Basic concepts of electromagnetism . . . . . . . . . . . . . . . . . . . . . 112.1.1. Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2. Electromagnetic fields in a medium . . . . . . . . . . . . . . . . . 122.1.3. Material dispersion: permittivity of a medium . . . . . . . . . . . 142.1.4. Complex refractive index . . . . . . . . . . . . . . . . . . . . . . . 162.1.5. Poynting vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.6. Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.7. Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.8. Absorbing medium . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2. General properties of the waveguide . . . . . . . . . . . . . . . . . . . . . 222.2.1. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2. Planar waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.3. Dispersion of a wave . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3.1. Group velocity dispersion of a waveguide . . . . . . . . . 262.2.4. Modal Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.5. TE and TM modes of a planar dielectric waveguide . . . . . . . . 282.2.6. Planar metal-dielectric interface: Plasmonic waveguide . . . . . . 29

2.2.6.1. TM modes of planar metal-dielectric interface . . . . . . 292.2.7. TE modes of planar metal-dielectric interface . . . . . . . . . . . 312.2.8. Cylindrical waveguides . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.8.1. Step index waveguide: dielectric core . . . . . . . . . . . 352.2.8.2. Step index waveguide: metal core . . . . . . . . . . . . . 362.2.8.3. Spiraling model for SPP modes on a wire . . . . . . . . 37

2.3. General properties of used fibers . . . . . . . . . . . . . . . . . . . . . . . 392.3.1. Photonic crystal fibers . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1.1. Solid core PCF . . . . . . . . . . . . . . . . . . . . . . . 40

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Contents

2.3.2. Modified step index fiber . . . . . . . . . . . . . . . . . . . . . . . 41

3. Basic theory for semiconductors 43

3.1. Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2. Band theory of semiconductors . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1. Electron in a periodic potential: Brillouin zones . . . . . . . . . . 473.2.2. Effective mass: concept of hole . . . . . . . . . . . . . . . . . . . 503.2.3. Carrier concentration of electrons and holes . . . . . . . . . . . . 513.2.4. Intrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . 533.2.5. Extrinsic semiconductors . . . . . . . . . . . . . . . . . . . . . . . 553.2.6. Mobility and drift of charge carriers . . . . . . . . . . . . . . . . . 57

3.3. Thermal properties of semiconductors . . . . . . . . . . . . . . . . . . . . 583.3.1. Thermal expansion coefficient . . . . . . . . . . . . . . . . . . . . 593.3.2. Effect of temperature on energy bandgap . . . . . . . . . . . . . . 593.3.3. Effect of temperature on carrier concentrations . . . . . . . . . . . 61

3.4. Optical properties of semiconductors . . . . . . . . . . . . . . . . . . . . 633.4.1. Optical absorption in semiconductors . . . . . . . . . . . . . . . . 63

3.4.1.1. Fundamental absorption process: intraband electronic ex-citations . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1.2. Absorption due to excitons . . . . . . . . . . . . . . . . 663.4.1.3. Absorption due to dopant, imperfection and impurity . . 673.4.1.4. Absorption due to intraband transitions and free carriers 673.4.1.5. Temperature dependence of optical absorption . . . . . . 673.4.1.6. Pressure dependence of optical absorption . . . . . . . . 683.4.1.7. Reflection spectra . . . . . . . . . . . . . . . . . . . . . 69

3.4.2. Recombination process . . . . . . . . . . . . . . . . . . . . . . . . 703.4.2.1. Radiative recombination process . . . . . . . . . . . . . 703.4.2.2. Nonradiative recombination process . . . . . . . . . . . . 723.4.2.3. Recombination rates . . . . . . . . . . . . . . . . . . . . 72

3.4.3. Photoconductivity in semiconductors: Intrinsic photoconductivity 74

4. Basic theory for filling 79

4.1. Contact angle and curvature of the meniscus of liquid . . . . . . . . . . . 804.2. Capillary effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3. Washburn’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5. Material properties 85

5.1. Dielectric: silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2. Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1. Germanium (Ge) . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3. Nobel metals: gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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Contents

6. Fabrication, characterization, measurement and simulation techniques 95

6.1. Fiber fabrication: stack and draw method . . . . . . . . . . . . . . . . . 956.1.1. Photonic crystal fiber . . . . . . . . . . . . . . . . . . . . . . . . . 956.1.2. Modified step index fiber . . . . . . . . . . . . . . . . . . . . . . . 102

6.2. Post-processing techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2.1. Flame based tapering rig . . . . . . . . . . . . . . . . . . . . . . . 1066.2.2. Splicer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2.3. Selective hole treatment . . . . . . . . . . . . . . . . . . . . . . . 108

6.3. Filling techniques in PCFs . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3.1. High temperature vacuum or pressure cell . . . . . . . . . . . . . 1126.3.2. Splicing technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.3. Direct drawing method . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4. Characterization techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4.1. Conductivity measurement . . . . . . . . . . . . . . . . . . . . . . 1216.4.2. Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.5. Optical measurement techniques . . . . . . . . . . . . . . . . . . . . . . . 1236.5.1. Optical transmission measurement . . . . . . . . . . . . . . . . . 1246.5.2. Photoconductivity measurement . . . . . . . . . . . . . . . . . . . 125

6.5.2.1. Side pumping . . . . . . . . . . . . . . . . . . . . . . . . 1266.5.2.2. Transmission mode . . . . . . . . . . . . . . . . . . . . . 127

6.6. Simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.6.1. Finite element method: JCMWave . . . . . . . . . . . . . . . . . 129

7. Results: germanium-filled structures 131

7.1. Material characterization of Ge-filled structures . . . . . . . . . . . . . . 1317.1.1. Conductivity measurements . . . . . . . . . . . . . . . . . . . . . 1317.1.2. Micro-Raman measurements . . . . . . . . . . . . . . . . . . . . . 132

7.2. Optical characterization of Ge-filled structures . . . . . . . . . . . . . . . 1337.2.1. Completely filled ESM PCF . . . . . . . . . . . . . . . . . . . . . 1337.2.2. Single hole filled ESM PCF . . . . . . . . . . . . . . . . . . . . . 134

8. Results: metal filled structures 147

8.1. Gold-filled modified step index fiber . . . . . . . . . . . . . . . . . . . . . 1478.1.1. Electrical conductivity measurements . . . . . . . . . . . . . . . . 1478.1.2. Optical transmission measurements . . . . . . . . . . . . . . . . . 148

8.2. Gold-filled polarization maintaining (PM) PCF . . . . . . . . . . . . . . 151

9. Device applications: germanium-filled structures 157

9.1. In-fiber temperature sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2. In-fiber photodetector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.2.1. Side pumping mode . . . . . . . . . . . . . . . . . . . . . . . . . . 1599.2.2. Transmission mode . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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Contents

9.3. High nonlinearity application of a germanium waveguide: theoretical analysis166

10.Conclusion and Outlook 169

A. Splicing parameters 173

B. Kramer-Kroenig’s relations 175

C. Instruments 177

viii

Abstract

The work reported in this thesis investigate the semiconductor- and metal- filled solid-core

photonic crystal fibers (PCFs). Using the techniques developed in this work, successful

filling of hollow channels of PCF with germanium (Ge) and gold (Au) has been achieved

by two different approaches. High optical quality gold or germanium single wires or

wire arrays with diameters as small as 100nm and aspect ratios larger than 10,000 were

fabricated successfully. The optical measurements clearly reveal excitation of Mie-type

resonances or spiraling plasmonic modes in such hybrid PCF structures.

Micro-Raman and conductivity measurements have indicated that the cm-long Ge-wires

have a high degree of crystallinity. The optical transmission spectra for a sample which

has a single Ge-wire placed adjacent to the core of an endlessly single-mode photonic crys-

tal fiber showed strong polarization dependent transmission losses in the visible region.

In the IR region, the anti-crossings between the glass-core mode and Mie-type resonances

in the high index Ge-wire creates a series of dips in the spectrum of the glass-core mode.

The measurements show a close agreement with the results of finite-element simulations.

The temperature dependence of the anti-crossing wavelengths suggests that a Ge-filled

structure can be used as a compact optical thermometer. In another experiment, a

Ge-filled MSIF was successfully used to measure the photocurrent in the Ge-wire.

The optical transmission spectra of the glass-core mode of directly drawn Au-filled mod-

ified step index fiber (MSIF) shows strong coupling of light from the glass-core mode

to SPP resonances on the Au-wire at specific wavelengths. The analytical and FEM

simulations match closely with the experimental findings. For a polarization maintaining

(PM) PCF, the transmission spectra of a sample with a single Au-wire adjacent to the

core also shows the excitation of plasmonic resonances. Near-field optical imaging of the

light emerging from such a Au-filled fiber confirms that the light from glass-core mode

is coupled to the surface plasmon modes on the wire. A spiraling plasmon model for the

propagation of SPP on wires is used for analytical calculations, which approximates the

dispersion of SPPs on wires very well down to wire diameters of just 100nm.

The successful fabrication and implementation of these novel structures is a step forward

to fabricate PCF based plasmonic and semiconductor in-fiber devices. These devices

can find possible applications in, ultra-low-threshold nonlinear optical devices, in-fiber

wavelength-dependent filters and polarizers, near-field tips for sub-wavelength scale imag-

ing and ion-trapping experiments.

Contents

Zusammenfassung

Diese Doktorarbeit beschäftigt sich mit halbleiter- und metallgefüllten Photonischen

Kristallfasern (engl. Photonic Crystal Fibers PCF). Das Füllen von PCF mit Germa-

nium (Ge) und Gold (Au) wurde mit zwei unterschiedlichen Verfahren realisiert, welche

im Rahmen dieser Doktorarbeit entwickelt wurden. Dabei können qualitativ hochwer-

tige Drähte (einzeln oder als Matrix) hergestellt werden mit Durchmessern größer 100

nm und einem Längen-/Durchmesserverhältnis > 10.000. Optische Messungen dieser hy-

briden PCF-Strukturen zeigen die Anregung von Mie-Resonanzen und sich auf helikalen

Trajektorien bewegenden plasmonischen Moden.

Mikro-Raman- und elektrische Leitfähigkeitsmessungen zeigten, dass die zentimeterlan-

gen Ge-Drähte eine weitgehend kristalline Morphologie besitzen. Für einen einzelnen Ge-

Draht direkt neben dem Kern einer einmodigen (engl. endlessly single-mode) PCF zeigen

optische Transmissionsmessungen starke, polarisationsabhängige Verluste im sichtbaren

Bereich des Spektrums. Im Infrarotbereich verursachen Anti-Crossing Punkte (zwischen

der Kernmode im Glas und Mie-artigen Resonanzen im Ge-Draht) eine Reihe von Min-

ima. Diese Ergebnisse sind in guter Übereinstimmung mit Simulationen, welche mit

Hilfe der Methode der finiten Elemente (FEM, engl. finite element method) ausgeführt

wurden. Da die Resonanzposition der Anti-Crossing Punkte temperaturabhängig ist,

kann die Ge-gefüllte Faser als ein kompaktes, optisches Thermometer benutzt werden.

In einem weiteren Experiment wurde in einer Ge-gefüllte, modifizierte Stufenindexfaser

(MSIF, engl. modified step-index fiber) ein Photostrom in dem Ge-Draht gemessen.

Optische Transmissionsspektren einer im Ziehturm gefertigten, Au-gefüllten MSIF zeigen

für bestimmte Wellenlängen eine starke Kopplung zwischen der Kernmode und den Plas-

monenresonanzen auf dem Golddraht. Analytische und FEM-Simulationen weisen sehr

gute Übereinstimmung mit den Messergebnissen auf. Für eine polarisationserhaltende

PCF mit einem einzelnen Au-Draht direkt neben dem Kern zeigen Transmissionsmes-

sungen ebenfalls eine Anregung plasmonischer Resonanzen. Optische Nahfeldmessungen

am Auskoppelende der Faser belegen, dass das Licht der Kernmode zum Oberflächen-

plasmon auf dem Draht koppelt. Ein Modell - basierend auf einem sich helixförmig auf

den Draht ausbreitenden Plasmon - wird für die theoretischen Berechnungen verwendet.

Dieses Modell spiegelt näherungsweise die Dispersion der Plasmonenmoden auf Drähten

bis zu einem Drahtradius von ca. 100 nm wieder.

Die Herstellung und Umsetzung dieser neuartigen Strukturen ist ein Schritt in Richtung

2

Contents

integrierter Faser-Optik basierend auf Metall- und Halbleiter-gefüllten PCF. Mögliche

Anwendungen sind nichtlineare optische Komponenten mit sehr niedriger Zerstörschwelle,

in die Faser integrierte (wellenlängenabhängige) Filter und Polarisatoren, Nahfeld-Spitzen

für Subwellenlängenoptik sowie Ionenfallen.

3

Contents

Key words

Photonic crystal fibers, semiconductors, germanium, gold wires, plasmonics, surface plas-

mon polariton, nanomaterials, in-fiber devices.

Key words

Photonische Kristallfasern, Halbleiter, Germanium, Golddrähte, Plasmonik, Oberflächen

Plasmon Polariton, Nanomaterialien, integrierte Faseroptik.

4

Acronyms

1D → one dimensional

2D → two-dimensional

3D → three-dimensional

Ag → silver

Au → gold

Ar → argon

CCD → charge coupled device

ESM → endlessly single mode

FEM → finite element simulation

Ga → gallium

He-Ne → helium neon

ID → inner diameter

IMPRS → International Max-Planck Research School

MSIF → modified step index fiber

NA → numerical aperture

OD → outer diameter

OSA → optical spectrum analyzer

PBG → photonic band gap guidance

PCF → photonic crystal fiber

PM → polarization maintaining

SEM → scanning electron microscope

SMF → single mode fiber

SNOM → scanning near field optical microscopy

SP → surface plasmon

5

Contents

SPP → surface plasmon polariton

SC → super continuum

TE → transverse electric

TIR → total internal reflection

TM → transverse magnetic

ZDW → zero dispersion wavelength

6

Chapter 1Introduction

Light-matter interactions provide a great insight in understanding the fundamental pro-

perties of matter and light and have been instrumental in the emergence of novel photonic

devices. Over the years, the interaction of light with semiconductors and metals has been

a highly researched area in photonics. One necessary requirement to study these inter-

action is the precise control over the coupling of light in matter.

Based on the previous studies of photonic band gaps [1] [2], the idea of two-dimensional

electromagnetic band gaps was proposed for fused silica in 1991, these frequency bands

are special frequencies that cannot pass through the designed material. Philip Russell

suggested the creation of structures with stop bands by fabricating SiO2 fibers with arrays

of microscopic hollow channels running along the fiber length. This concept of trapping

light was experimentally implemented in 1996 [3]. These fibers are termed as photonic

crystal fibers (PCFs). The ability of these special fibers to overcome the limitations of

conventional fibers have opened new possibilities in the area of photonics e.g. control

over dispersion of the fiber, high nonlinearity, low loss guidance in air and interaction of

light with gases, metals and semiconductors [4] [5] [6] [7].

The history of semiconductor materials dates back to 1833 when Michael Faraday ob-

served the extraordinary case of increase in the electrical conductivity of silver sulfide

crystals with temperature [8]. It was opposite to the behavior observed in copper and

other metals at that time. This new phenomena lead to the invention of semiconductor

materials. Over the years, their unique electrical and optical properties have been uti-

lized with numerous breakthroughs in inventing new electronic or optoelectronic devices

as well improving the performances of many other devices. The existence of one of the

7

1. Introduction

most common semiconductor, germanium, was predicted by Mendeleev but it was discov-

ered in 1886 by Winkler [9]. Germanium was also the first material on which transistor

effect was observed by John Bardeen and Walter Brattain in 1949 [10]. In early 1960s,

germanium had been the main material for the development of bipolar transistors before

being overcome by silicon. The optical properties of germanium are still widely used in

infrared optics [11] [12]. Additionally, the high index of refraction of germanium is used

in doping of the conventional optical fibers with GeO2 to achieve an index guidance. The

doping of GeO2 in optical fiber core has also been used to enhance the nonlinearity of

the silica fiber.

In an attempt to realize the opportunity provided by PCF to fabricate in-fiber devices,

the filling of empty holes in the PCFs by silicon and germanium was reported by Sazio

et al. [13], relying on depositing germanium and silicon in the holes with the help

of chemical precursors. In an another approach, filling of pure germanium (without any

chemical precursors) in the hollow channels of solid-core PCFs has been reported recently

by Tyagi et al. [5].

When light interacts with metal surfaces, it excites electrons which can form propa-

gating waves called surface plasmon polaritons (SPPs). They are collective vibrations

of the electron gas propagating on the surface of metals and confines light in the near

vicinity of metal-dielectric interfaces. This confinement leads to a strong enhancement

of the electromagnetic fields, which are highly sensitive to surface conditions. The first

experimental and theoretical study of surface plasmons can be related to the work of

Sommerfeld [14] and Mie [15]. The SPPs have found applications in sensing (due to

their sensitivity to the surface), nano-scale optics, photonic circuits and also in realizing

optical materials [16] [17] [18]. The ability of SPPs to confine light in small volumes

has also lead to the invention of several new surface analysis techniques e.g. Scanning

near field optical microscopy (SNOM). Current research in the area of plasmonics uti-

lizes plasmonic excitation on planar structures or nano particles. One of the reason for

utilizing these geometries is the availability of fabrication techniques. Such techniques

(lithography and sputtering) were originally developed for nanoscale electronic devices

and have been optimized over the years for the electronics industry.

The cylindrical geometry in the form of metal wires has also proved be an efficient way to

investigate SPPs and can be considered as a promising alternative to dielectric waveguides

in highly integrated optical devices. SPPs excitation on metal nanowires have also found

useful applications in surface-enhanced Raman spectroscopy (SERS) and waveguiding in

8

terahertz frequency range [19].

Using the waveguiding properties of PCFs and allowing long interaction lengths for the

light (guided in a core) to couple with metal wires (in adjacent holes near the core), an

unique possibility exists for exciting the SPP modes efficiently on the metal wire. The

excitation of SPPs on metal wire arrays in solid-core PCFs has recently been successfully

reported by Schmidt et al.[7] and Lee et al. [6]. These samples were fabricated by a new

technique in which empty holes of the PCF structure were filled with gold and silver.

The transmission spectrum and optical imaging of the end face confirmed that the SPP

resonances were excited on metal wires. Theoretical investigation of these structures lead

to a new perspective to understand the propagation of SPPs on wires, which proposes

that SPP modes spiral around the circumference of the metal wires [7] [20]

Some important questions raised during the work can summarize the basic motivation of

the thesis;

• Is it possible to fill PCFs with semiconductor materials and metals ?

• What are the effects of semiconductors and metals wires on the trans-

mission characteristics of the PCFs glass-core mode ?

• What are the possible the fiber structures (other than PCF) which can

be fabricated to investigate the light-matter interactions?

• What kind of in-fiber devices can be realized using these novel struc-

tures?

The following chapters are not only an attempt to find answers to above mentioned ques-

tions but also to explore new directions in the area of light-metal and light-semiconductor

interactions in PCFs. The work reported in this thesis has useful contributions from Dr.

Markus Schmidt, Patrick Uebel and Howard Lee.

structure of the thesis

This thesis contains ten chapters. After the first introductory chapter, the second chapter

provides the basic information on the modes in a dielectric and metal waveguide. This

chapter also offers a brief overview of the spiraling SPP model proposed by Schmidt et

al. [7]. Chapter three describes some of the electrical and optical properties of semicon-

9

1. Introduction

ductors which are useful in the context of this work. Chapter four gives a brief insight

into the filling process of liquids, which is useful in fabricating germanium or gold -filled

PCFs. Chapter five introduces the properties of different materials used in this thesis.

Chapter six explains the different techniques that are used in fabricating and analyzing

the optical properties of Ge-filled or Au-filled structures. Chapter seven deals with the

results of germanium-filled structures, while chapter eight contains the results of Au-filled

structures. Some in-fiber devices are proposed in chapter nine. Chapter ten concludes

this work and provides the future outlook in relation to this work.

10

Chapter 2Basic theory for waveguiding

2.1. Basic concepts of electromagnetism

2.1.1. Maxwell’s equations

In classical electro-dynamics and optics, all electro-magnetic phenomena can be fully

understood by considering light to be made of electric and magnetic fields. These fields

are general solutions of the four Maxwell’s equations (MEs) proposed by James Maxwell

in 1865 [21]. Based on several experimental findings, these equations form the basis of

the entire electro-magnetic field theory. Maxwell’s equations combine Gauss’s law for

electric and magnetic fields, Faraday’s law of induction and Ampere’s circuit law. All

four equations can be written as a set of linear partial differential equations1,

∇ · D = ρ (2.1a)

∇ · B = 0 (2.1b)

∇× E = − ∂

∂tB (2.1c)

∇× H = J +∂

∂tD (2.1d)

with dielectric displacement field D, magnetic induction B, electric field E, magnetic

field H,charge density ρ and current density J. The macroscopic form of these equations

use the charge and current densities which are averaged over microscopically large, but

1In SI units.

11

2. Basic theory for waveguiding

macroscopically small volumes. Macroscopically averaged fields vary smoothly in space

and are mathematically well behaved.

2.1.2. Electromagnetic fields in a medium

The electric field and magnetic field are fundamental field vectors, while D and B could

be considered as the effective response of the medium under investigation. Therefore,

one can relate E with D and B with H by taking into account the properties of the

medium. Under the influence of an external field, charged particles present inside the

medium can give cause three important phenomenas, i.e. conduction, electrical polari-

zation, or magnetization. A particular material can possess all three properties but they

can be classified as conductor, dielectric or magnetic material depending on whichever

phenomena is predominant. A detailed introduction of the behavior of electromagnetic

waves inside the medium can be found for example in [22].

For conductors, free electrons move under the influence of an external field with an

average velocity proportional to the applied field strength. The current density J relates

with the electric field E as,

J = σE (2.2)

where σ is the electrical conductivity of the medium. Equation 2.2 is Ohm’s law. Po-

larization in a material refers to the phenomenon of the creation and net alignment of

electric dipoles, which are formed by displacing the centroid of the electron cloud of the

atoms along the direction of the applied electric field. In a macroscopic volume, a net

dipole moment P (polarization vector) is the measure of this displacement of the charge

cloud. Hence, the effective field response D in the presence of the external field E can

be represented as

D = ǫ0E + P (2.3)

In a similar manner, the effect of magnetization can also be taken into account in the

presence of a magnetic field through a relationship between B and H

B = µ0 (H + M) (2.4)

It is quite evident from these equations that in the absence of the dielectric or magnetic

materials (i.e. vacuum) the polarization P and magnetization M terms are zero. Pola-

rization and magnetization are proportional to the applied electric and magnetic field,

12


respectively and can be expressed as

P = ǫ0χeE (2.5a)

M = ǫ0χmH (2.5b)

The χe and χm are dimension-less quantities named as the electric susceptibility and the

magnetic susceptibility.

For complete understanding of the dependence of polarization P on electric field E and

magnetization M on magnetic field H, one needs to expand equation 2.5a and equation

2.5b as power series. During the expansion, the spatial dependence of the E and material

properties as well as the frequency ω dependence of the material properties must be taken

into account. The expansion can be expressed as

Pi(ω) = ǫ0

[

χ(1)e(ij)

(ω)Ej + χ(2)e(ijk)

(ω)EjEk + ......]

(2.6a)

Mi(ω) = µ0

[

χ(1)m(ij)

(ω)Hj + χ(2)m(ijk)

(ω)HjHk + ......]

(2.6b)

where Ei,Ej, Ek and Hi, Hj, Hk are spatial components of the electric and magnetic

fields giving rise to polarization Pi and magnetization Mi.

The χe and χm with n = 1,2,3 . . . are electric susceptibility and magnetic susceptibility

tensors. The first order term in equation 2.6a and 2.6b gives rise to linear dependence

of polarization P on E and magnetization M on H. For a homogeneous and isotropic

medium (χe, ǫ, χm and µ converts to constant scalar quantities), combining the first

order term of 2.6a and 2.6b with 2.5a and 2.5b for linear regime, material equations for

D and B can b expressed as

D = ǫ0(1 + χe(ω))E (2.7a)

= ǫ0ǫ(ω)E (2.7b)

B = µ0(1 + χm(ω))H (2.7c)

= µ0µ(ω)H (2.7d)

These equations are useful in the regime of linear optics. The linear dependence of field

vectors is appropriate when the electric field strength is lower than the strength of fields

which exits at the atomic level, e.g. coulomb fields. In the context of the thesis most

of the discussion is in linear regime unless specified. The electrical susceptibility χe,

13


permittivity ǫ, magnetic susceptibility χm and permeability µ are complex (except for

the vacuum), indicating that polarization P and displacement field D do not always

remain in phase with the electric field E.

2.1.3. Material dispersion: permittivity of a medium

The dependence of the electric permittivity on the excitation frequency is called the

material dispersion. It can be interpreted as the response function of a material that

depends on the frequency of the field. This frequency dependence arises because the

medium does not get polarized instantaneously under the application of electric field.

The response of the medium, i.e. permittivity ǫ of the medium, is represented by a

complex quantity (ǫr+iǫi) as it allows to specify the phase and magnitude of the response.

The real part of the permittivity ǫr is related to the energy stored by the medium while

imaginary part ǫi is related to the dissipation (loss) of energy.

There are different models which can explain the permittivity of the medium. The

Lorentz oscillator model, which is based on the fact that electrons are displaced due

an electric field and undergo periodic oscillations, these oscillations are damped due to

the radiation of energy due to the movement of electrons. According to this model,

permittivity of the medium is given by,

ǫ(ω) = 1 +ω2p

ω20 − ω2 − iγω

(2.8)

where

ωp ≡Ne2

mǫ0(2.9)

Here, γ is the damping constant for oscillating molecules and ωp is the plasma frequency.

The Plasma frequency is determined by the number of dipoles in the medium and is the

resonance frequency of the medium’s pure plasma state in which the electrons oscillates

against the positive ion cores. The real and imaginary parts of equation 2.8 are plotted

in figure 2.1.2 The real part can be divided into three different regions:

• ω ≤ (ω20 - γω0)

1/2: in this region, permittivity increases with frequency and the

material is said to be in a normal dispersion regime. The lower frequency limit of

2The values of ωp, ω0 and γ used in this plot does not represent any real material, they are chosen toreflect the behavior of the permittivity well.

14


0.0

0.5

1.0

1.5

2.0

Re( )ε

Im( )ε

Perm

ittivity

ε

Frequency

( +ω γ0

2ω0)

1/2( -ω γ0

2ω0)

1/2

ω0

Figure 2.1.: Real and imaginary part of the complex permittivity around a resonance frequencyω0 calculated with the Lorentz oscillator model. The black dashed line corresponds to resonancefrequency ω0. The green and blue dashed line correspond to the higher and lower frequencylimit of Re(ǫ).

this region is

ǫ(ω) ≈ 1 +

(

ωp

ω0

)2

(2.10)

• (ω20 - γω0)

1/2 ≤ ω ≤ (ω20 + γω0)

1/2: the permittivity falls from its maximum value

at ω = (ω20 - γω0)

1/2

Re(ǫ)min = 1 +ω2p

2γω0 − γ2(2.11)

to its minimum value at ω = (ω20 + γω0)

1/2

Re(ǫ)min = 1−ω2p

2γω0 − γ2(2.12)

This is called the anomalous behavior of the material, which occurs near the reso-

nance frequency.

• (ω20 + γω0)

1/2 ≤ ω: the permittivity again increases with increasing frequency and

the material shows normal dispersion. In the high frequency limit, the permittivity

can be expressed as

ǫ(ω) ≈ 1−(

ωp

ω0

)2

(2.13)

15


The imaginary part of the permittivity peaks strongly at the resonance frequency ω0.

The peak height increases with the damping constant. For frequencies far away from ω0,

the imaginary part approaches zero.

For metals, equation 2.13 represents the permittivity of the metal as suggested in Drude

Model, which is based on the assumption that Im(ǫ(ω)) is zero.

The Lorentz oscillator model must be extended for real materials since they have several

resonance frequencies in a given frequency range. If fk is the number of electrons within

a given atom (molecule) having resonance frequencies of ωk and damping coefficients of

γk, the permittivity can be written as

ǫ(ω) = 1 +e2

2ǫ0m

∑ Nkfkω2k − ω2 − iγkω

(2.14)

here summation is over all k’s, which represents the number of resonances, while fk is the

strength of the oscillator for each resonance. [23]. For the high quality silica, in which

the damping term in negligible for visible and near IR region, the permittivity is real

and equation 2.14 reduces to Sellmeier equation [24]. The detailed discussion is given in

section 5.1.

2.1.4. Complex refractive index

Another key parameter which characterizes the optical properties of the medium is the

complex refractive index n. It is in terms of permittivity and permeability

n =√ǫµ = nr + iκ (2.15)

Here, nr refers to the ratio of the speed of light in vacuum to the speed of light in

a medium. The imaginary part κ is called the extinction coefficient, which refers to

the absorption of a medium. For most of the cases in this work the permeability µ is

considered as unity unless stated otherwise. The real part of the dielectric function ǫr

and the imaginary part of dielectric function ǫi is related to n and k as

ǫr = n2r − κ2 (2.16a)

ǫi = 2nκ (2.16b)

16


2.1.5. Poynting vector

In the context of this work, the flow of energy associated with an electromagnetic field

guiding through the medium (linear) is very important and can the defined in terms of

total energy density as

u ≡ 1

2

(

ǫ0ǫ|E|2 + ǫ0ǫ|H|2)

(2.17)

This total energy density must satisfy the following energy conservation law [25],

∂u

∂t+∇ · S = −J · E (2.18)

Equation 2.18 is called the Poynting’s theorem, in which the term S is known as Poynting

vector and can be defined as the S = E × H 3, which is the electromagnetic energy flowing

per unit area normal the direction of E and H. The total power associated with the field

can be calculated by integrating S over a closed surface area. According to Poynting’s

theorem, the change of electromagnetic energy with time in a given volume plus the

amount of energy flowing out through the boundaries of that volume, is negative of the

mechanical and thermal work done by the electric field on the sources within the volume.

In real medium the dispersion of the medium must also be taken into account.

2.1.6. Wave equations

The electric and magnetic fields are coupled in Maxwell’s equations but can be decoupled

under some assumptions. In a linear medium, using equation 2.7c and (ǫ and µ are a

scalar and constant quantity) MEs can be transformed in

∇ · E =ρ

ǫ(2.19a)

∇ · H = 0 (2.19b)

∇× E = −µ0µ∂

∂tH (2.19c)

∇× H = σµ0µE + µ0µǫ0ǫ∂

∂tE (2.19d)

For all dielectric materials relevant to this work ρ and σ are zero i.e. there are no free

charge or current densities. Under these assumptions, taking the curl of equation 2.19c

and then using 2.19d and 2.19b, the equation for the electric field vector can be expressed

3In case of complex fields S = 1/2 E × H*.

17


as

∇2E = µµ0ǫǫ0∂2

∂t2E + µµ0σ

∂

∂tE (2.20)

In a similar manner, the equation for magnetic field can also be obtained,

∇2H = µµ0ǫǫ0∂2

∂t2H + µµ0σ

∂

∂tE (2.21)

Equation 2.20 and 2.21 are called vector wave equations for a linear medium. On com-

paring these equation with mechanical counterparts one can suggest that the quantity

µǫ represents the inverse of the square of the speed for the electromagnetic waves in the

medium. Therefore, the velocity c of the EM waves in the medium is determined by the

permittivity ǫ and µ of the medium,

c =1√

ǫǫ0µ0µ(2.22)

If the wave functions E(r, t) or H(r, t) are harmonically oscillating in time with an angular

frequency ω, then

E(r, t) = E(r)e−iωt (2.23a)

H(r, t) = H(r)e−iωt (2.23b)

Substituting these values in in wave equation 2.20 gives

∇2E = µǫǫ0µ0ω2E − iµ0µσωE (2.24a)

∇2H = µǫǫ0µ0ω2H − iµ0µσωH (2.24b)

Equation 2.24 can also be written as

∇2E + k2E = 0 (2.25a)

∇2H + k2H = 0 (2.25b)

Here,

k2 = µǫǫ0µ0ω2 − iµµ0σω (2.26)

18


The term k is the complex wave number and is the absolute value of the complex wave

vector. The complex wave vector k can expressed as

k = kr + ia (2.27)

Here, kr is the real part of the complex wave vector and imaginary part a is called the

attenuation vector. The absolute value of the imaginary part of the complex wave vector

(absolute value of a) is responsible for the attenuation for the wave as it propagates in

the medium. The amplitude of the real and imaginary parts of the complex wave vector

can be expressed as

kr = ω

√

ǫ0ǫµµ0

2

√

√

√

√

√

1 +

(

σ

ǫ0ǫω

)2

+ 1 (2.28a)

a = ω

√

ǫ0ǫµµ0

2

√

√

√

√

√

1 +

(

σ

ǫ0ǫω

)2

− 1 (2.28b)

where,

ω =k√

µǫǫ0µ0

=kc0√µǫ

= ck (2.29)

The equations 2.25 are called Helmholtz equations for electromagnetic waves. Using

THE method of separation of variables, the general solution of these equations can be

expressed as

E(r) = E0ei(k·r+φe) (2.30a)

H(r) = H0ei(k·r+φm) (2.30b)

For each solution of a wave propagating along k, there exists another solution for a counter

propagating wave in -k. Therefore, the complete solutions for the wave equations can be

expressed as

E(r, t) = E0ei(k·r−ωt+φe) = E0e

iφeei(k·r−ωt) (2.31a)

H(r, t) = H0ei(k·r−ωt+φm) = H0e

iφmei(k·r−ωt) (2.31b)

This clearly indicates that, in a general case the amplitude part of 2.31a, i.e E0eiφe , and

2.31b, i.e H0eiφe , can be complex quantities and may not be always remain in phase. The

terms φe and φm represents the constant phase attached with the electric and magnetic

19


fields amplitudes. These phase terms are independent of time and space coordinates

and essentially works as a constant offset between the electric and magnetic fields. The

relative phase difference between electric and magnetic field is important during the

calculation of the Poynting vector. In this work, the real part E0 of the complex amplitude

E0eiφe is considered as the amplitude of the electric field unless mentioned otherwise. The

same discussion holds true for the magnetic field. Therefore, the equation 2.31a and 2.31b

can we written as,

E(r, t) = E0ei(k·r−ωt) (2.32a)

H(r, t) = H0ei(k·r−ωt) (2.32b)

As the wave equations are linear in E and H, the solutions of the these equations can be

expressed as the superpositions of the basis functions.

2.1.7. Plane waves

One of the most common forms of the solutions are plane waves. A plane wave is

characterized by a constant phase i.e at any given time the phase of the wave is constant

in a plane perpendicular to the propagation direction of the wave. If a plane wave is

propagating in a vacuum in z direction, the complex amplitude of the wave E0 has two

independent components ((E0) = (Ex(x, y, z), Ey(x, y, z), 0)).

A plane wave is linearly polarized, if the phase difference between the two components is

an integer multiple of π or one of the components is 0, then the electric field oscillates in

the same plane. For a finite phase difference between these components the electric field

vector generally creates an ellipse which gives rise to elliptically polarized light. In the

case |Ex| = |Ey|, the phase difference is π/2 and the light is called circularly polarized

(the electric field vector tip creates a circle). It is important to note that this condition

holds only for a non-absorbing media where a = 0. The magnitude of real and imaginary

parts of the complex refractive index (n = nr + i κ) and complex wave vector (k = kr

+ i a) are related to each other as,

kr2 − a2 = (nr

2 − κ2)ω2

c02(2.33a)

kr · a = nrκω2

c02(2.33b)

20


For a transparent media where a = 0 and κ = 0, the frequency dependence of the wave

number can be expressed as,

k(ω) =ω

cn(ω) =

ω

c

√

ǫ(ω)µ(ω) (2.34)

With this definition the velocity of the wavefront can be expressed as,

vp =ω

k=

c0ǫ(ω)µ(ω)

=c0n(ω)

(2.35)

It is called the phase velocity of the wave. It is clear that the phase velocity of the

medium is a function of refractive index, which is function of frequency. This frequency

dependence of phase velocity is the main cause of the dispersion of a wave in a medium.

2.1.8. Absorbing medium

By substituting equation 2.27 in 2.36 and rearranging the terms, it can be shown that the

attenuation vector is responsible for decreasing the field amplitude during propagation.

The electric field vector can be expressed as

E(r, t) = E0e−a·rei(kr·r−ωt) (2.36)

The distance required to reduce the amplitude of the field by 1/e is called the penetration

depth of the material (for metals, it is called the skin depth). It is the quantity that shows

how far the wave penetrates into the medium.

In a complex notation the complex wave number can also be expressed as k = |k| eiφ

|k| =√

k2r + a2 = ω√ǫ0ǫµ0µ

√

1 +

(

σ

ǫ0ǫω

)2

+ 1 (2.37a)

φ = tan−1(a/kr) (2.37b)

Using the complex notations of the field amplitudes for electric and magnetic fields, it

can be shown that φe − φm = φ. Therefore, for an absorbing medium the electric and

magnetic fields are not in phase and this phase difference is related to the complex wave

vector.

21


2.2. General properties of the waveguide

A waveguide is physical geometry in which electromagnetic waves are confined and can

propagate. The term optical waveguide refers to waveguides which are used to transmit

electromagnetic waves at optical frequencies (visible and near infra-red frequencies). In

general, a waveguide consists of a core medium surrounded by a cladding medium. The

propagating electromagnetic waves are confined in the core of the waveguide. The analysis

of a waveguide is associated with the study of the guided modes in the waveguide. A

complete description of optical waveguide can be found in various textbooks of optical

waveguide [26] [27]. A mode of a waveguide is the eigen state of the system and also

a mathematical solution of a wave equation. To understand the wave guiding behavior

of the electromagnetic fields at the interfaces of the medium must be analyzed. This is

achieved by applying the boundary conditions

2.2.1. Boundary conditions

Electromagnetic field vectors must satisfy boundary conditions at the interface of the two

linear mediums. Figure 2.3 shows a schematic representation of a surface dividing two

mediums. The boundary conditions must be satisfied across the surface. The explicit

form of the discontinuity can be deduced in the form of following general relations [27],

E H

D B1 1

1 1

E H

D B2 2

2 2

Medium ‘1’

Medium ‘2’

K ρ

n

Figure 2.2.: Schematic diagram showing a surface with current density K and charge density ρdivides the two mediums. n is the unit vector normal to the surface.

(D1 −D2) · n = ρ (2.38a)

(B1 −B2) · n = 0 (2.38b)

(E1 − E2)× n = 0 (2.38c)

(H1 −H2)× n = K × n (2.38d)

22


Here, K is the free surface current density 4, n is the unit vector perpendicular to the

surface and subscript 1 and 2 indicates the two mediums. These equations are general

boundary conditions for electrodynamics. In context of this work, equation 2.38c and

equation 2.38d are most important, which are related to tangential components of the

field. For most of the cases related to this work, surface current density is zero at the

interface. Therefore, the tangential magnetic field components are also continuous. For

the case of a real metal, in the absence of any external electric field electrons are in free

random motion, which results into a net zero surface current density.

2.2.2. Planar waveguide

A planar waveguide is characterized by its refractive index or dielectric function distri-

bution, which in the discussed case depends only on one dimension, i.e. n2 = n2(x) or ǫ

= ǫ(x). For the case of symmetric waveguide, the MEs reduce to two independent sets of

equations giving rise to two set of solutions. Figure 2.3 shows the schematic design of a

x

y

z

ε1

ε2

Material ‘1’

Material ‘2’

Material ‘1’ε1

2a

Figure 2.3.: Schematic design of planar waveguide. A layer of material ’2’ is sandwiched betweentwo layers of material ’1’.

planar waveguide. The waveguide is a three layered structure of two different materials

with dielectric function ǫ1 and ǫ2. The layer with material ǫ2 is sandwiched between two

layers of material with ǫ1. The thickness of the middle layer is 2a on the x- axis and the

slabs have infinite dimensions in y- and z- directions. The direction of propagation is

defined along z axis. As the waveguide is infinite in the y direction the electromagnetic

fields of the waveguide modes depends on x and z only. In this case, the electric and

magnetic field components can be expressed as,

E(x, z) = E(x)eiβze−iωt (2.39a)

H(x, z) = H(x)eiβze−iωt (2.39b)

4It is related to the free current density J.

23


Here, β is the propagation constant (z-component of the wave vector k). The main task

is to find the values of β for different modes and their respective field distributions, for a

particular frequency ω. Using the Helmholtz equation (equation 2.25), the field vectors

of the three regions of the waveguide can be expressed as

[

∂2

∂x+ k0

(

ǫ1 − n2eff

)

]

ψ1 = 0 (2.40a)[

∂2

∂x+ k0

(

ǫ2 − n2eff

)

]

ψ2 = 0 (2.40b)[

∂2

∂x+ k0

(

ǫ1 − n2eff

)

]

ψ3 = 0 (2.40c)

The term neff is called the effective refractive index of the mode and is related to

the propagation constant β as, neff ≈ β/k0. The scalar ψ is proportional to each field

component of the electric and magnetic fields in each of the three regions, which means

that each field component must satisfy these equations. The subscript 1, 2, 3 represents

three different regions of the waveguide. Now, three different cases can arise for Re(neff )2,

• Re(neff )2 > Re(ǫ1) > Re(ǫ2); in this case, using the boundary conditions, it can

be shown that the fields increases exponentially, which can not be considered as a

physical solution.

• Re(ǫ1) < Re(neff )2 < Re(ǫ2); the field components are exponentially decaying in

region 1 and 3 and shows sinusoidal behavior in region 2. These modes are called

guided modes.

• Re(neff )2 < Re(ǫ1) < Re(ǫ2); the field components are sinusoidal in all three regions

and the corresponding modes modes are called radiation modes.

Substituting the expression for the field values from equation 2.39 in Maxwell’s equations,

the field components for a planar waveguide can be expressed as,

Ex = ineffλ0

2π(ǫ− n2eff )

∂Ez

∂x(2.41a)

Ey = −i λ02π(ǫ− n2

eff )

1

ǫ0c0

∂Hz

∂x(2.41b)

Hx = ineffλ0

2π(ǫ− n2eff )

∂Hz

∂x(2.41c)

Ey = iλ0

2π(ǫ− n2eff )

ǫ0c0ǫ∂Ez

∂x(2.41d)

24


Therefore, the transverse components of the fields can be calculated using only the field

components in the propagation direction (z-axis) and also by determining the dispersion

relation of propagation constant β or effective index neff . The boundary conditions,

explained in section 2.2.1, necessitates that the tangential component of the electric and

magnetic fields must be continuous. This leads to two independent set of equations [28],

• Continuity of the tangential magnetic field at the surface leads to the fact that the

tangential component of the electric field must vanish. In this case, only Ey, Hx

and Hy components are non zero. As the electric field components lie in the plane

that is perpendicular to the direction of propagation (z-axis), these types of mode

are termed as transverse electric (TE) modes.

• If the boundary condition of the continuity of electric field is applied, then the

tangential component of the magnetic field must vanish. For this case, nonzero

components are Ez, Ex and Hy. As the nonzero magnetic field component Hy lies

in the plane perpendicular to the direction of propagation, these types of modes

are termed as transverse magnetic (TM) modes.

In a more general case, Ez and Hz are non zero (for cylindrical waveguides, section 2.2.8).

In this case the modes are called hybrid modes with designation HE and EH modes5.

As discussed in equation 2.41, all the transverse field components can be obtained from

Ez and Hz. Helmholtz equation for Ez can be expressed as6,

[

∂2

∂x+(

k0ǫ− β2)

]

Ez(x) = 0 (2.42)

The solution for Ez for the guided modes in the structure can be expressed as

Ez(x) = Aeγ1x x ≥ d

Ez(x) = Beγ2x + Ce−γ2x 0 < x < d

Ez(x) = Deγ3x x ≤ d

(2.43)

5Field is EH if Ez has major contribution and HE if Hz has dominant contribution.6In a similar manner the Hz also satisfies Helmholtz equation.

25


Inserting 2.43 in 2.42, expression for γ1, γ2 and γ3 can be obtained.

γ21 = k20(n2eff − ǫ1) → γ1 = k0

√

(n2eff − ǫ1) (2.44a)

γ22 = k20(ǫ2 − n2eff ) → γ2 = ik0

√

(n2eff − ǫ2) (2.44b)

γ23 = k20(n2eff − ǫ1) → γ3 = k0

√

(n2eff − ǫ1) (2.44c)

These equations implies that γ1 ≡ γ2. For simplicity a parameter α can be defined as α

= k0√

(n2eff − ǫ), i.e. γ2 = iα.

2.2.3. Dispersion of a wave

In general, dispersion corresponds to the phenomena that the phase velocity of light

depends on the frequency of light. A guided wave in a medium travels with a surface

of constant phase, which is normal to its propagation direction z. The velocity of this

phase front along the z axis is the phase velocity of the wave,

vp =ω

β(2.45)

In case of TE, TM or HE wave β is not a linear function of ω, leading to a phase velocity

which is a function of ω. This type of propagation is called dispersive propagation. The

variation of propagation constant β (or neff ) with frequency ω is referred to as dispersion

diagram in this work.

2.2.3.1. Group velocity dispersion of a waveguide

For a pulse propagating in a waveguide, the velocity at which this wavepacket (pulse)

travels along the propagating direction is called the group velocity and is defined as

vg =dω

dβ(2.46)

The different spectral components associated with a pulse (wavepacket) experiences diffe-

rent speeds c/n(ω) leading to a pulse dispersion. The dispersion of the pulse is expressed

26


in terms of group velocity dispersion GVD. The GVD parameter is defined as [29]

D(λ) = −2πc

λ2d2β

dω2(2.47)

The variation of D with λ has a interesting feature that its value can be zero at specific

wavelengths and is positive at longer wavelengths (multiple zeros can also be obtained).

The wavelength at which D = 0 is called zero dispersion wavelength (ZDW). The GVD

plays a very important role in pulse propagation mechanisms. At wavelengths shorter

than λZDW , wave packets of long wavelength travel faster than the wave packets of short

wavelength and it is called the normal dispersion regime. For wavelengths longer than

λZDW , it is the other way around and is called the anomalous dispersion regime. [29].

2.2.4. Modal Parameters

Based on the definitions of γ2 and γ1, dimensionless parameter U and W can be defined

for material ’2’ and ’1’, respectively. Using the definition that β = neffk0, U and W can

be expressed in terms of the refractive indices of the materials

U = ak20(n22 − β2) (2.48a)

W = ak20(n21 − β2) (2.48b)

where a is half the width of the core of the waveguide. Using 2.48 waveguide parameter

V can be defined as

V 2 = U2 +W 2 (2.49a)

V =2πa

λ

(

n22 − n2

1

)

(2.49b)

The V -parameter of the waveguide is a very important parameter for defining the number

of guided modes. For a given value of V , the propagation of bound modes must satisfy

the following conditions,

0 ≤ U < V and 0 < W ≤ V (2.50)

It is important to mention the following two important properties for the modes in the

waveguide

27


• Modal cut off wavelength : The lower limit of the wavelength of a mode which can

be obtained for Re(neff ) = k0n1. It gives U = V and W = 0. It is important to

note that the fundamental mode of the waveguide has no cut-off.

• Number of modes: The number of modes for a planar waveguide can be defined in

terms of V parameters; N ≈ Int(4V /π), where Int corresponds to the largest value

of integer below (4V /π).

2.2.5. TE and TM modes of a planar dielectric waveguide

The dispersion relation for TE and TM modes can be obtained by applying the boundary

conditions on the field solutions of the Helmholtz equation for Ez and Hz. Normalized

propagation constant β/k0 versus normalized frequency V (which is related to frequency)

is plotted as a dispersion diagram for first few TE and TM modes in figure 2.4. This plot

is obtained for the case ǫ2/ǫ1 = 2.5.

0 0.5 1.0 1.5 2.0 2.5 3.0 3.51.0

1.2

1.1

1.3

1.4

1.5

V

β/k0 TE0

TM0

TE1

TM1

π/2

Figure 2.4.: Normalized propagation constant β/k0 versus V for TE0, TM0, TE1 and TM1

modes for planar dielectric waveguide. The green dot corresponds to V = π/2 [27].

• Number of modes: N ≈ Int(4V/π), If V ≤ π/2, then only the fundamental mode

will be guided through the waveguide. The green dot in the figure 2.4 corresponds

28


to the V = π/2.

• Cut-off wavelength: The cutoff wavelength for a planar waveguide is given by V -

parameter as Vc = mπ/2, where m is the mode order and m = 0, 1, 2 ,3 . . . . It

shows that the fundamental mode has no cutoff and therefore will always be guided

in the core. If we design the waveguide such that 0 < V < π/2, then the waveguide

is single mode. The maximum value of U for a mode is given by (j + 1)π/2

2.2.6. Planar metal-dielectric interface: Plasmonic waveguide

A plasmon is an oscillation of the electrons in a metal with respect to the positive

charged ions [30]. If these oscillations are confined to the surface, they are referred

to surface plasmons (SPs), which can interact strongly with the external light leading

to the formation of polariton. A polariton is a quasi particle resulting from a strong

coupling of electromagnetic wave with an electric or magnetic dipole carrying excitation.

In conclusion, a surface plasmon polariton (SPP) is an oscillation of the electron plasma

near the surface of the medium, that propagates along the interface. In the context of

a metal-dielectric interface, the surface plasmon polariton (SPP) are the surface waves

which have an evanescent field in the metal and dielectric.7 As discussed in the previous

section, the modes of a waveguide can be divided into TE and TM modes.

2.2.6.1. TM modes of planar metal-dielectric interface

In order to obtain a guided mode at a metal-dielectric interface, the fields must be

evanescent in both the dielectric and the metal. Ez in this case can be expressed as

Ez(x) = Ae−γDx x ≥ d

Ez(x) = BeγMx x < 0(2.51)

The need to have evanescent field in the metal and dielectric given that the Re(γD) and

Re(γM) must be greater than zero. Inserting these equations in Helmholtz equation gives

γ2D = β2 − ǫDk20 → γD = k0

√

(n2eff − ǫD) (2.52a)

γ2M = β2 − ǫMk20 → γM = k0

√

(n2eff − ǫM) (2.52b)

7A complete description about plasmons can be found in [31].

29


Using the boundary condition for Ez and Hy at the interface gives A = B and

γMγD

= −ǫMǫD

(2.53)

Equation 2.53 is a very important relation in context of the modes at a metal-dielectric

interface. For a guided mode, γD and γM are positive and also the permittivity ǫD is

positive. For metals the ǫM is always negative for frequencies above the plasma frequency.

It indicates that the equation 2.53 holds true for metals and TM plasmon modes exists

on a metal dielectric interface. The dispersion of a planar plasmon can be expressed as

neff =

√

ǫDǫMǫD + ǫM

(2.54)

Using equation 2.13 for the frequency of a perfect metal(Drude model), the dispersion

relation for a planar SPP can be written as

neff =

√

ǫD(λ2p − λ2)

(1 + ǫD)λ2p − λ2(2.55)

Here, λp is the plasma wavelength, which corresponds to the plasma frequency. Figure 2.5

shows the dispersion relation of planar surface plasmon for silver. The plasma wavelength

of the silver was assumed to be 140nm. There are three regions in this plot;

150 200 250 300 3500

1

2

3

4

5

ne

ff

Wavelength (nm)

silica

SPP

λp λsp

air

A B C

Figure 2.5.: Dispersion (red curve) of a planar SPP mode for a lossless metal. The λp (140nmfor silver) is the plasma frequency. The black curve indicates the dispersion of silica.

30


• λ < λp, region A: the effective index of the SP modes lies below the silica line. As

silver is transparent in this region and it can be shown that electromagnetic fields

oscillate in this region. This region is called radiative plasmon polariton.

• λp ≤ λ ≤ λsp, region B: The real part of the effective index is zero and the effective

index is purely imaginary, which means no SPP exists. This is only the case for a

lossless medium. In the case of a real metals SPPs can propagate in this region but

with extremely high loss.

• λ ≥ λsp, region C: For the case of λ ≫ λsp, the effective index is close to the light

line, which means a plasmon mode behaves like a plane wave.

2.2.7. TE modes of planar metal-dielectric interface

Now let us consider the case of TE modes. As the fields inside the metal and dielectric

must be evanescent, the Hz can be expressed as

Hz(x) = Ce−δDx x ≥ d

Hz(x) = DeδMx x < 0(2.56)

Inserting these equations in Helmholtz equation gives,

δD = k0

√

(n2eff − ǫD) (2.57a)

δM = k0

√

(n2eff − ǫM) (2.57b)

Now, using the boundary conditions for Ey and Hz at the interface, it can be shown

that C = D and

C(δD + δM) = 0 (2.58)

As, the real part of δD and δM are greater than zero, equation 2.58 can only be satisfied

when C = 0. This indicates that for a planar metal-dielectric interface the SPP does not

exist for TE modes. [31].

2.2.8. Cylindrical waveguides

One of the most common forms of a waveguide geometry is the cylindrical step index

configuration, e.g., conventional step index fiber. The complete analysis of a cylindrical

31


waveguide can be found in various text books [27] [26].

Let us consider a simple step index configuration of a waveguide in cylindrical geometry

as shown in figure 2.6. The host material is defined by its permittivity ǫD and the core

εM

εD

Material ‘D’

Material ‘M’

a

θ

z

x

y

r

Figure 2.6.: Schematic design of the cylindrical step index waveguide. The material D withpermittivity ǫD is the cladding and material M is the core of the waveguide with permittivityǫM . The light is propagating along the z axis and a is the radius of the core.

material is defined is permittivity ǫM with the radius a. 8 If the waveguide properties

are invariant along z-direction, then, the electric and magnetic fields can be expressed

as,

E(r, θ, z) = E(r, θ)eiβze−iωt (2.59a)

H(r, θ, z) = H(r, θ)eiβze−iωt (2.59b)

The symmetry for θ requires that after a complete cycle of 2π, the field components

must satisfy the boundary conditions that E(r, θ) = E(r, θ + 2π), which indicate that

the r and θ dependence of the fields can also be expressed as,

E(r, θ) = E(r)ei(±m)θ (2.60a)

H(r, θ) = H(r)ei(±m)θ (2.60b)

where, m= 1, 2, 3, . . . .9 Similar to the planar waveguide case, the transverse components

8The material ’M’ can be metal, semiconductor or another dielectric.9Here, −m indicates a anticlockwise rotation but in this work positive m is used.

32


of the field vectors can be expressed in terms of the Ez and Hz

Eθ = − λ02π(ǫ− n2

eff )

[

mneff

rEz +

i

c0ǫ0

∂

∂rHz

]

(2.61a)

Er =λ0

2π(ǫ− n2eff )

[

ineff∂

∂rEz −

m

c0ǫ0rHz

]

(2.61b)

Hθ =λ0

2π(ǫ− n2eff )

[

ic0ǫ0ǫ∂

∂rEz −

mneff

rHz

]

(2.61c)

Hr =λ0

2π(ǫ− n2eff )

[

mc0ǫ0ǫ

rEz + ineff

∂

∂rHz

]

(2.61d)

This indicates that the solutions of Ez and Hz are essential to calculate the transverse

field components. Ez and Hz must satisfy the Helmholtz equation, which is same for Ez

and Hz.10. Therefore, both Ez and Hz can be expressed in terms of a common function

ψ as

Ez(r) ∝ ψ(r) (2.62a)

Hz(r) ∝ ψ(r) (2.62b)

Inserting the function ψ(r) into the Helmholtz equation for a cylindrical coordinate

system and then multiplying by r2 gives a Bessel differential equation,

[

r2∂2

∂r2+ r

∂

∂r

]

ψ(r) +[

r2(

ǫk20 − β2)

−m2]

ψ(r) = 0 (2.63)

Now for the case, Re(ǫk20 − β2) > 0, the solution of equation 2.63 can be expressed in

terms of Bessel functions of first and second kind as [32] [26]

ψ = C1Jm(k0kMr) + C2Ym(k0kMr) (2.64)

For Re(ǫk20 − β2) ≤ 0, the solution of equation 2.63 can be expressed as,

ψ = C3Im(k0kDr) + C4Km(k0kDr) (2.65)

where kD ≡√

(n2eff − ǫ) while C3 and C4 are the constants. The Im(u) and Km(u) are

the modified Bessel functions of first and second kind. As discussed in section 2.2.2 for

a planar waveguide case, for a guided mode in the core region (material M), the fields

10Ez and Hz differ only by a constant of proportionality.

33


must exponentially decay in the dielectric material (material D). Using the properties

of Bessel functions, for the case r > a, the solution is given by 2.65, with C3 = 0. If

coefficient C4 is chosen in such a way that fields are normalized at r = a, then

ψD(r) =Km(k0kDr)

Km(k0kDa)(2.66)

For the modes inside the core material the fields must be finite at r = 0, which means

C2 must be zero and

ψM(r) =Jm(k0kMr)

Jm(k0kMa)(2.67)

The Ez and Hz can now be expressed as

Ez(r, θ, z) = ψ(r)eimθeiβz (2.68a)

Hz(r, θ, z) = Aψ(r)eimθeiβz (2.68b)

A is the proportionality constant and

ψ(r) =

ψM(r), r ≤ a

ψD(r), r > a(2.69)

Using the condition that the tangential components of E and H must be continuous

at the boundary, the dispersion relation for a cylindrical waveguide of material with

dielectric function ǫM in a host with dielectric function ǫD can be expressed as

ǫMa2k4Dψ

21 + ǫDa

2k4Mψ22 + (ǫD + ǫM)a2k2Mk

2Dψ1ψ1 −m2n2

eff (ǫM − ǫD) = 0 (2.70)

with,

ψ1 =∂

∂rψM at r = a (2.71a)

ψ2 =∂

∂rψD at r = a (2.71b)

Equation 2.70 is a general dispersion relation which can solved numerically for neff at a

fixed wavelength.

34


2.2.8.1. Step index waveguide: dielectric core

A simple example of the step index fiber with a dielectric core is a conventional step

index fiber. A complete description of the modal properties of step index fiber can be

found in various text books, e.g. [26] [33]. Some important conclusions of the modal

analysis are briefly mentioned here. Similar to a planar case, the dispersion relation for

a step index waveguide can be obtained, as shown in figure 2.7. Some important modal

HE11

EH11

TE01

HE12

TM01

β/k0

V

0 1 2 3 4 5 6

Normalized frequency

n2

n1

Figure 2.7.: Dispersion relation for some lowest order modes (HE11, TE01, TM01, EH11 andHE12) for a step index waveguide with dielectric core.

properties are discussed below,

• Number of modes: The number of modes for a step index waveguide is given by N

= Int(V 2/2) [26].

• Cut off wavelength: By calculating the dispersion curves, the condition for a fun-

damental mode guidance in a step index fiber is 0 < V < 2.405, which means that

the waveguide is single mode if V lies in this range.

• The modes of cylindrical waveguide are hybrid modes, identified by HEml and

EHml modes.11 Here, subscript m corresponds to the mode order and l denotes

the root order of the eigenvalue equation of dispersion. The fundamental mode of

a cylindrical waveguide is HE11 mode as it has the highest value of β for a given

mode order and has no cutoff.11For azimuthally symmetric structures some modes can be expressed as TE0l or TM0l.

35


2.2.8.2. Step index waveguide: metal core

It is important to note that metal has a negative real part of the dielectric function. Figure

2.8 shows the dispersion relation for the first five modes on a silver wire of diameter ≈1000nm embedded in silica. It is clear from figure 2.8 that modes with an order ≥ 2 have

400 600 800 1000 1200

1.44

1.50

1.56

1.62

Re

(n

)e

ff

Wavelength (nm)

silica

0

1234

a = 500nm

Figure 2.8.: Dispersion relation for different modes of a silver wire (diameter ≈ 1µm) embeddedin silica. The black line is the dispersion of silica and numbers indicate the SPP mode orders.

a cutoff, indicating that these modes are not guided beyond these wavelengths. Figure

2.8 further indicates that these modes extend beyond their crossings to the silica line.12

This interesting behavior is attributed to the large imaginary part of the ǫM . As the

wavelength increases, the modes spreads out in the dielectric causing a decrease of the

loss of the mode, which reduces the overlap between the mode and the metal wire.

The dispersion of both 0th and 1st order stay above that of silica silica line. It can

be shown from the loss calculations that although the loss of 1st and 0th order modes

decreases with wavelength, the 1st order mode shows a relatively lower loss than 0th.

In case of a metal film with few 10s of nm thickness and sandwiched between two identical

dielectrics, the attenuation of surface plasmons with greatly reduced for one of the modes.

These “long range” modes consist of anti-phased surface plasmons on opposite sides of the

metal layer, with a field zero in the center. Due to this zero field intensity in the middle,

12For a loss-less case the modal cut off is exactly at the silica line because the dispersion of the modesare always above the silica line.

36


these modes can propagate longer distances in comparison to a planar symmetric mode.

The 0th and 1th order modes of the metal wires is found to be equivalent to symmetric

and antisymmetric modes of a planar metal film sandwiched in between two dielectric

layers.

2.2.8.3. Spiraling model for SPP modes on a wire

In a simple model, SPPs modes on wire can be treated as planar SPP modes spiraling

around the wire on a helical path. This model has been published by Schmidt et al.[7].

The wave vector of the spiraling SPP can be decomposed into axial wave vector β (which

kspp

βkθ

εM

εD

Dielectric

Metal

a

θ

x

y

r

Figure 2.9.: Schematic of a spiraling SPP which follows a helical path around a Au-wire em-bedded in silica.

corresponds to the propagation constant) and an azimuthal component kθ as,

|kspp|2 = |kθ|2 + β2 (2.72)

Such spiraling SPs are in fact inhomogeneous “plane waves” with phase-fronts that are

oriented at an angle to the axis. Therefore, the wave vector of the plasmon must satisfy

the dispersion for the planar SPPs (equation 2.55). To achieve constructive interference

in order to form a mode, the azimuthal wavelength (λθ) must be equal to an integer

multiple of the circumference of wire (2πa), i.e. mλθ = 2πa. This gives the expression

for azimuthal number kθ = m/a. BY combining these expressions, the dispersion of a

37


spiraling SPP can be obtained as

neff =

√

ǫDǫMǫD + ǫM

−(

m

ak0

)2

(2.73)

When comparing the results for the spiraling plasmon model of SPP on a wire to the

exact solutions, it is observed that a good agreement can only be achieved by reducing

the mode order in by 1. This could be attributed to the additional phase which SPP

gains when propagating on a helical path.13 Hence, the dispersion relation of spiraling

SPP on a metal wire can be expressed as,

neff =

√

ǫDǫMǫD + ǫM

−(

m− 1

ak0

)2

(2.74)

which is valid for modes m ≥ 1. Figure 2.10 shows the comparison of the exact analytical

400 600 800 1000 1200

1.44

1.50

1.56

1.62

Wavelength (nm)

silica

1234

a = 500 nm

Re

(n

)e

ff

Figure 2.10.: Comparison of the exact calculations (solid lines) with the spiraling plasmonmodel (dashed lines) for the dispersion of a silver wire. The numbers on each line indicatethe SPP mode orders.

calculations and spiraling plasmon model for the dispersion of a silver wire. The disper-

sion calculations shows that the results from the spiraling plasmon model predicts better

agreement with the exact calculations for larger wire diameters. For smaller wires, their

is discrepancy which can attributed to that fact at larger curvature of the nanowire, the

local planar approximation became increasingly invalid.13This is called Berryś phase [34].

38

2.3. General properties of used fibers


2.3.1. Photonic crystal fibers

Photonic crystals are periodically structured electromagnetic media, whose periodicity

length scale is of the order of the used wavelength of light. This medium can be considered

as the electromagnetic analogue of the crystalline atomic lattice. Like their electronic

counterpart, these photonic crystals possess photonic band gaps14, i.e. ranges of fre-

quency in which light cannot propagate. The study of these structures is governed by

the Bloch theorem. Intentionally added defects give rise to localized electromagnetic

states e.g. linear waveguides or point like cavities. The study of electromagnetic wave

propagation in a periodic media goes back to 1887, when Lord Rayleigh studied the re-

flective properties of a crystalline mineral with periodic twinning planes. This type of

material is essentially a one-dimensional photonic crystal. Lord Rayleigh identified that

these minerals have a narrow band gaps prohibiting the light propagation through the

planes and these band gap are angle dependent. These bands are stop-bands which forms

the basic concept of preventing light to propagate in all directions i.e. photonic band

gaps.

Over the years, this concept has been widely studied to manipulate light propagation in

one, two or three dimensions, giving rise to the concept of one, two or three dimensional

photonic crystals. In 1991, Philip Russell proposed that refractive index requirements

for PBG formation in two dimensions are greatly relaxed if the propagation of light is

predominantly in a third direction, similar to that of an optical fiber. The idea of Philip

Russell to trap light in a hollow core by means of a two-dimensional photonic crystal of

microscopic air capillaries running along the length of glass fiber was successfully demon-

strated by Knight et al. in 1996 [3]. These special fibers were termed photonic crystal

fibers (PCFs) as the guidance mechanism of such fibers is based on a two-dimensional

array of holes (photonic crystal) in silica15. A review article related to the PCFs and

their possible applications was published by Philip Russell in 2006 [4].

PCFs can be widely classified into two categories depending on the design of the core,

i.e. solid-core PCFs and hollow-core PCFs. In the context of this work, solid-core PCFs

are important and are briefly discussed in the following section 2.3.1.1.

14In a crystal lattice these band gaps are called electronic band gaps as explained in section 3.2.15PCFs made of other materials has also been studied. As the this work is based on air-silica PCFs so

the term PCF in this work refers to air-silica PCFs unless specified.

39


2.3.1.1. Solid core PCF

In a solid core PCF, a silica core is surrounded by a transverse periodic array of air holes

which are typically arranged in a hexagonal symmetry16. The guidance mechanism of

these fibers is based on effective index guidance, which is also the concept of guidance in

a conventional step index fiber. The periodic arrangement of the arrays of holes reduces

the effective index of the cladding and creates a positive index difference between silica

core and cladding [35] [36] [37]. The solid-core PCFs can be further classified in to several

other types, depending on their structural properties and applications.

Endlessly single mode (ESM) PCF: The endlessly single mode (ESM) solid-core

PCFs are made of pure silica and show low loss single mode guidance of the core at all

wavelengths. For a mode with proportion constant β, the condition for guidance in core

is

knco > β > βFSM (2.75)

Here, k is the free space wave vector of light and βFSM is the propagation constant

of the fundamental space filling mode (FSM) . The FSM is the mode with the highest

proportion vector in cladding, i.e. fundamental mode of the cladding. It is related to the

effective refractive index of the cladding as

neff = βFSMk (2.76)

If compared to a conventional step index fiber, βFSM is equal to kncl. Similar to the

conventional step index fiber, if the PCF is designed (diameter to pitch ratio d/Λ) in

such a way that Veff is less then a critical value of V parameter, a single mode behavior

can be achieved for all wavelengths. This value of d/Λ is 0.45 for a silica-air PCF [35].

The behavior of single mode guidance in ESM PCF can be understood by viewing the

array of holes as modal filters. The fundamental mode in the glass-core has a transverse

effective wavelength as λgeff ≈ 4Λ, which makes it unable to “squeeze through” the glass

between the holes, which has width of Λ - d. If the d/λ is of correct dimensions (0.45),

all the higher order modes are able to escape through the cladding as their transverse

effective wavelength is small, thus achieving single mode guidance[4].

One of the important applications of ESM fiber is the generation so called supercontinuum

(SC) light from pico-second or femto-second laser pulses. While a high-power pulse

16Non birefringent fibers.

40


propagates along the PCF, its frequency spectrum broadens due to a range of nonlinear

effects such as four-wave mixing. This broadband light source is considered to be the

most important applications of PCFs and has various applications across many areas of

science.

Polarization maintaining PCF: Another important type of a solid-core design is the

polarization maintaining PM PCF, in which the six-fold symmetry of the ESM core is

broken and a two-fold rotational symmetry is generated [38]. This type of fiber is used

to achieve different dispersions for two orthogonal polarizations.

If the core and cladding index difference is negative, then guidance can be achieved only

due to the photonic band gap of the cladding, as found in the case of hollow core [4] [36]

[39].

2.3.2. Modified step index fiber

Beside PCFs, another type of fiber structure to investigate the light matter interactions

can be obtained by modifying the structure of a conventional step index fiber. This new

structure is called “modified step index fiber” (MSIF) in this work. A small air hole can be

introduced adjacent to the GeO2-doped silica core at a desired spacing. This hole is then

filled with metals/semiconductors and the transmission through the core is measured to

investigate the optical properties of such a structure. Figure 2.11 shows the schematic

core

air-hole

Figure 2.11.: Schematic design of the modified step index fiber (MSIF).

design of a modified step index fiber (MSIF). The central core region consists of GeO2-

doped silica. The doping level of the GeO2 is ≈ 16 mol.% in pure silica, which increases

the refractive index of the core with respect to the surrouding silica. The dispersion of

16 mol.% GeO2-doped silica and pure fused silica is shown in figure 2.12. The variation

41


of the refractive index of silica with wavelength (dispersion) is obtained from Sellmeier’s

equation (section 5.1) and for GeO2-doped silica, it is obtained using published data from

Fleming [40].

400 800 1200 1600

1.44

1.46

1.48

1.50R

efr

active

in

de

x

Wavelength (nm)

GeO -doped silica core2

Fused silica

Figure 2.12.: Refractive index variation of silica and 16 mol.% GeO2-doped silica withwavelength.

It is clear from figure 2.12 that the doping of GeO2 increases the refractive index of

doped silica, which creates a positive refractive index difference between the core and

cladding of MSIF and allows the guidance of light due to total internal reflection. It

is also important to design the overall structure in such a way that the core has single

mode guidance for the wavelengths of interest. As discussed previously, for single mode

behavior at any wavelength, the V -parameter at that wavelength must lie below 2.405.

In an empty step index fiber, the introduction of an air hole does not affect the guidance

of a MSIF. This situation changes when the empty hole is filled with a high index material

like gold or germanium.

42

Chapter 3Basic theory for semiconductors

A material can be defined as a semiconductor material if (i) its electrical resistivity lies

in between electrical resistivity of typical metals and insulators i.e. in the range of 10−3

and 109 Ω− cm (ii) it has negative temperature coefficient of resistance (some semicon-

ductors e.g. PbS also has positive temperature coefficient) (iii) its electrical conductivity

depends on impurity content (e.g. doping, temperature, excess charge injection and op-

tical excitation). These factors can affect the electrical conductivity by several orders of

magnitude. This offers a unique opportunity for these materials to have potential appli-

cation in making many electronic and optoelectronic devices for generation and detection

of electromagnetic fields. The semiconductor materials can be classified on the basis of

their electronic band structure, chemical composition, crystalline structure or electronic

properties and their applications. In order to describe these materials it is important to

know their inter-atomic bonding configurations, structural properties and imperfections

in the material.

Some important characteristics of semiconductors are discussed in brief in this section.

For detailed information various textbooks can be referred [41] [42] [43].

3.1. Crystal structure

Based on their structural order, semiconductor materials can be classified as crystalline

and amorphous. Crystalline semiconductors are further classified in to two categories i.e.

single crystalline and polycrystalline. Although majority of the semiconductors used in

43

3. Basic theory for semiconductors

electronics are single crystalline yet there are some applications based on polycrystalline

and amorphous materials.

A crystal is characterized by a well defined arrangement of atoms in lattice. Most of the

semiconductors have diamond like (e.g. Silicon and Germanium) or zincblende like (e.g.

Gallium arsenide) lattice structures. These lattice structures have tetrahedral phases

which means that each atom is surrounded by a set of four nearest ’neighbor’ atoms,

that lie at the corners of the tetrahedral. Figure 3.1 shows the schematic arrangement of

a

Figure 3.1.: Diamond crystal structure with lattice spacing a.

atoms for a diamond crystal lattice. Germanium and silicon also have similar arrangement

of atoms in crystal lattice space.

A convenient method of defining a plane in a direct crystal lattice is Miller indices [43].

It is also possible to define the points in terms of reciprocal lattice vectors (momentum

space or k-space representation). Reciprocal lattice space is some time also referred as

wave vector space. The important relation between reciprocal (R) and direct (G) lattice

vectors can be expressed as

G · R = 2π × Integer (3.1)

Therefore each vector of reciprocal lattice is normal to a set of planes in the direct lattice.

The primitive cell in the reciprocal lattice is represented by the Wigner-Seitz cell. It is

constructed by drawing perpendicular bisector planes in the reciprocal lattice from the

center (of the three dimensional space) to the nearest equivalent reciprocal lattice sites.

The Wigner-Seitz cell in reciprocal lattice is also referred to as the first Brillouin zone

44

3.2. Band theory of semiconductors

(The detailed description of the concept of Brillouin zones is in section 3.2). Figure 3.2

K W

XU

L

Kz

Ky

Kx

Λ

Δ

∑

Г

Figure 3.2.: First Brillouin zone (Wigner-Seitz cell) of diamond crystal lattice. Important sym-metry points and lines are also indicated. Points of symmetry inside the zone is in Greek lettersand it is in terms of Roman letters for the surface.

shows a typical example of Wigner-Seitz cell (first Brillouin zone) for a diamond lattice

structure. It is important to note that standard notation for points of symmetry inside

the zone is in Greek letters and points of symmetry on the surface is in terms of Roman

letters . Thus, Γ represents the origin of the reciprocal space (i.e. k = 0); Λ represents

a direction such as [111] and L denotes the zone end along that direction. The concept

of the reciprocal lattice is very important to understand the band structure of the solids,

semiconductors in particular.


The basic properties of the semiconductor materials are analyzed by understanding the

motion of the electrons through its periodic crystal lattice structure, where electrons can

be considered as moving in a periodic crystal potential V (r) which has the periodicity of

the lattice. The wave nature of an electron allows us to express its motion in the form of

the Schroedinger equation. If we assume that the properties of the semiconductors does

not vary with time, then the time-independent Schroedinger equation can be expressed

45


as[

− h2

2m2∇2 + V (r)

]

ψ(r,k) = E(k)ψ(r,k) (3.2)

here, ψ(r,k) represents the wave function of an electron for a time-independent case i.e. a

stationary wavefunction. E represents the total energy of the electron and V(r) represents

the potential in a direct lattice space. Due to periodicity of the lattice, potential can be

expressed as V (r) = V (r + R), i.e potential is also periodic with the periodicity of the

lattice. For free electrons V (r) = 0.

One of the most important way to investigate a periodic structure is the Bloch theorem.

According to the Bloch theorem, solutions of equation 3.2 is of the form of Bloch func-

tions which are the product of a plane wave (exp i(k · r)) and a function Ub(r). This

function Ub(r) has same periodicity as periodic potential,i.e Ub(r) = Ub(r + R). Using

this property of Bloch theorem, complete solution of equation 3.2 can be expressed as,

ψ(r,k) = exp (ik · r)Ub(r) (3.3)

It is clear from the above discussion that ψ(r,k) = ψ(r + R,k). It is also necessary to

note that k · R is multiple of 2π. This property of Bloch function helps to visualize the

E-k relationship (band structure) in a reciprocal lattice space.

The physical meaning of ψ is that |ψ|2 dxdydz represents the probability of finding an

electron in a volume dxdydz in the vicinity of position (x, y, z). Solution of this equation

provides ψ and E as the eigen-function and the eigenvalue respectively.

It can also be shown from the Bloch theorem that the energy E(k) is periodic in reciprocal

lattice. For a given band, to label the energy uniquely, it is sufficient to use only k’s in

first Brillouin zone of reciprocal lattice. The standard convention is to use the Wigner-

Seitz cell in the reciprocal lattice. This cell is defined as the first Brillouin zone. Thus,

it is clear that we can reduce any momentum k in the reciprocal space to a point inside

the Brillouin zone. Using this property, E-k relationship of a crystal structure can be

shown by folding the bands in first Brillouin zone. This representation is called the

reduced zone representation of the band diagram. The E-k relationship of a particular

structure provides the most important information related to the band structure of that

semiconductor.

46


3.2.1. Electron in a periodic potential: Brillouin zones

For a free electron (in absence of any potential (V = 0)) solution of the equation 3.2

gives the allowed energy values which are essentially continuous

E =~2

2me

(

k2x + k2y + k2z)

(3.4)

here kx, ky and kz are three components of the wave vector of an electron. Equation 3.4

indicates that in the absence of potential V all energy values are allowed for electron.

According to Kronig-Penney model, electron motion in a one dimensional periodic crys-

tal lattice can be approximated to the motion of an electron in a periodic square well

potential. This model summarizes that in the presence of periodic potential the available

energies of an electron are modified and it can only possess energies within certain energy

bands and allowed energies are separated by the regions of forbidden energies.

The boundaries of allowed bands are at k = nπ/a, where k is wave vector of the electron,

n is the integer and a is the lattice spacing, this indicates the presence of discontinuity at

these values of k. These conditions are similar to Bragg reflection rules, which means that

the electron state with k = nπ/a can be described as a standing wave and the electron is

not allowed to propagate through the crystal lattice. Under this condition the equation

3.4 can be expressed as

(

2meE/~2)1/2

E1/2 = k + n(2π)/a (3.5)

which reflects a parabola is repeated periodically with n(2π)/a.

From the above discussion it is quite clear that E is a periodic function of k with peri-

odicity 2π/a. We know that for an electron in a periodic lattice, discontinuity of energy

occurs at the boundary of the allowed band when k = nπ/a, where n = ±1,±2,±3 . . . .

Deviation from the parabolic behavior of curves occurs at the band edges of k, which

means that an electron in a periodic lattice behaves similar to a free electron except

in the vicinity of k = nπ/a. Figure 3.3 shows the extended zone representation of E-k

relation for an electron in periodic potential.

Using the translation invariant property of the Bloch functions, the energy band structure

of an electron can also be represented in reduced zone scheme where the bands are folded

into the first zone. The shape of the Brillouin zone is determined by the lattice structure

47


0

3rd2nd 1st 2nd3rd

E

k

Free electronmodel

AB

AD

AB

FB

FD

FB

-π/a-2π/a-3π/a π/a 2π/a 3π/a

Figure 3.3.: Extended zone representation for E − k relation of Kronig-Penney model (nearlyfree electron model). Black dotted curve shows the E-k relation of a free electron. Solid curveshows the modifications of the parabolic E(k) dependence for the free electrons at the bandedges corresponding to k = π/a First three Brillouin zones are also indicated.(FB: Forbiddenband, AB: Allowed band)

(a) (b)

Figure 3.4.: E − k band diagram characterizing conduction and valance band for germaniumand silicon in reduced zone scheme. (a) Germanium (b) Silicon .

48


and its size depends on the lattice parameters. Figure 3.4 shows the energy band diagram

of two important semiconductors i.e. silicon and germanium in the reduced zone scheme.

Introduction of the spin can lead to degeneracy of the bands as shown in case of silicon

and germanium. For any semiconductor there is forbidden energy range in which no

allowed state exists. This is called the energy gap. The allowed energy regions above and

below the region are called the conduction band and the valance band. The separation

between the lowest conduction band and the highest valance band is called the bandgap

or the minimum energy gap Eg.

Indirectconductionband

Directconductionband

Valancebands

Heavy hole(HH) band

Light hole(LH) band

Split-offbandk = 0

E

k

Directbandgap Indirect

bandgap

+

ke kh

Electronremoved

Figure 3.5.: Schematic diagram representing direct bandgap and indirect bandgap semiconduc-tors. Different curvatures in valance band edges can led to light hole (LH) and heavy hole(HH)bands. Figure also illustrates the concept of hole as a missing electron from valance band. Thewavevector of hole is that of a missing electron.

Figure 3.5 shows the schematic representation of band diagram near the band edges for

any semiconductor. The location of the minimum energy gap plays a very crucial role

in the operation of semiconductor devices. The most important observation from the

band structure of silicon or germanium and GaAs is the location of the minimum of

49


the conduction band with respect to the location of the maximum of the valance band.

When the minimum of conduction band and the maximum of valance band occurs at the

same point in k-space, the semiconductor is called as direct band gap semiconductor such

as GaAs. In other cases, semiconductor is called indirect band gap semiconductor(e.g.

silicon and germanium.)

3.2.2. Effective mass: concept of hole

Periodic potential in crystal lattice not only modifies the electronic band structure but

it also modifies the electronic mass. The effective mass can be expressed in the terms of

curvature of the energy band as1

m∗=

1

~2

d2E

dk2(3.6)

The effective mass of an electron is inversely proportional to the curvature of the electronic

band. This indicates that the effective mass of an electron is determined by the E-k

diagram from which it is clear (i) effective mass is positive near the bottom of the band

and negative near the top of the band (ii) the effective mass is also independent of energy

near the top and bottom of the bands. The electron having a negative mass near the top

of the band is referred as a hole. Figure 3.5 also shows that the wave vector of a missing

electron is essentially equal in magnitude to the wave vector of a hole.

In presence of the electric field a negative mass with negative charge behaves as a positive

mass with positive charge. Thus, the holes are referred as charge carriers having positive

charge and positive mass. This implies that in presence of the electric field the holes

and electrons will travel in opposite direction in real space. The inverse dependence

of the effective mass on the curvature of the electronic band also indicates that the

greater curvature leads to small effective mass i.e light electrons (LE) or holes (LH) and

vice versa. Also the curvature of the bands are different for different crystallographic

directions which means that the effective mass of the electron is different for different

points on k-axis. In the valance band, presence of spin-orbit interaction also lead to the

splitting of bands at certain k value. These split bands are referred to as as split-off

bands as shown in the figure 3.5.

50


3.2.3. Carrier concentration of electrons and holes

Knowledge of carrier concentrations and the energy distributions is essential to under-

stand the electronic and optical properties of the semiconductors. To calculate the carrier

concentration, the knowledge of the density of states (DOS) and the probability of carrier

occupancy of the state at energy E is necessary. Both electrons and holes are the Fermi

particles, thus the occupation statistics of these carriers are described by the Fermi-Dirac

statistics. This means that the probability of an electron, occupying a state at energy E

is described by the Fermi-Dirac distribution function

fn(E) =1

exp [(E − EF ) /kBT ] + 1(3.7)

where T is the temperature, kB is the Boltzmann constant and the term EF is called as

the Fermi energy (also referred as the Fermi level). The Fermi energy is defined as the

energy at which the probability of the occupation is 1/2 i.e f(E) = 1/2. Similarly, the

probability of a hole occupying a energy state E at a given temperature is given by

fp(E) = 1− fn(E) =1

exp [(EF − E) /kBT ] + 1(3.8)

In a semiconductor, only those electrons are mobile which lies above the conduction

band level Ec. The total number of electrons in a semiconductor lying at energy band

dE is proportional to the (i) number of states per unit volume per unit energy i.e density

of states g(E) over the interval range of E and E + dE (ii) probability of an electron

occupying a state at energy E i.e f(E). The total number of electrons in the conduction

band can be obtained by evaluating the integral of the product of g(E) and f(E) over

entire energy range of conduction band. As the Fermi level is much below the top of

the conduction band the upper limit can be replaced by the infinity and thus the total

number of the electrons in the conduction band is

n =

∫

∞

Ec

gn (E) fn (E) dE (3.9)

The density of states which is the number of states dN lying between k and k + dk can

be obtained by dividing the spherical volume between k and k + dk i.e 4πk2dk by the

volume of the unit cell occupied by one state with specific k. The factor of two must also

be included in this expression to accommodate the spin effect, hence the total expression

51


becomes,

dN = Vk2dk

π2(3.10)

For parabolic bands (effective mass is constant), the k and kdk can be expressed in as

k =[

2m∗

e (E − Ec) /~2]1/2

(3.11a)

kdk = m∗

edE/~2 (3.11b)

Putting the expression of k and kdk, the expression for the DOS can be written as,

gn(E) = 4π(

2m∗

e/h2)3/2

(E − Ec)1/2 (3.12)

At room temperature kBT is about 0.026 eV . If E−EF ≫ kBT , the Fermi-Dirac function

can be approximated to

fn(E) =1

exp [(E − EF ) /kBT ] + 1∼= exp [(E − EF ) /kBT ] (3.13)

Hence the concentration of electrons in the conduction band can be expressed by substi-

tuting equation 3.12 in 3.9

n = 2

(

m∗

e2kBT

2π~2

)

F1/2 (ηn) (3.14)

where, F1/2 (ηn) is the Fermi integral and ηn = (EF − Ec) /kBT . Here

Nc = 2

(

m∗

e2kBT

2π~2

)

(3.15)

is referred as the effective density of states for the conduction band. Similarly the ex-

pressions for the concentration of holes in the valance band is ,

p = 2

(

m∗

h2kBT

2π~2

)

F1/2 (ηp) (3.16)

where F1/2 (ηp) is the Fermi integral and ηp = (Ev − EF ) /kBT . Here

Nv = 2

(

m∗

h2kBT

2π~2

)

(3.17)

52


is referred as the effective density of states for the valance band. Now if η ≤ −3, F1/2(η)

can be approximated as exp(η). It means, for η ≤ −3 one have Ec − EF ≥ 3kBT and

EF − Ev ≥ 3kBT , so the electron and hole concentrations can be expressed as

n = Nc exp [(EF − Ec) /kBT ] (3.18a)

p = Nv exp [(Ev − EF ) /kBT ] (3.18b)

The condition η ≤ −3 implies that the Fermi level position in energy gap is much

greater than 3kBT from the band edges. These types of semiconductors are called the

non-degenerate semiconductors. The semiconductors for which the Fermi level is within

3kBT range of band edges (minimum of conduction band and maximum of valance band)

are called degenerate semiconductors.

3.2.4. Intrinsic semiconductors

Semiconductor materials without impurities are called intrinsic semiconductors. In case

of intrinsic semiconductor, thermal activation of an electron from valance band to con-

duction band produces an electron in the conduction band and a hole in the valance

band. This type of electronic transition across the band gap produces an electron-hole

pair (not necessary a bound electron-hole pair). After a random motion through the

lattice, the excited electron can encounter a hole and subsequently goes into a recom-

bination process. The generation and recombination are continuous processes and the

average life time is referred as life time of the charge carrier. During these processes, the

concentration of electrons in the conduction and holes in the valance band are equal i.e.

n = p = ni (3.19)

Figure 3.6 schematically shows the variation of the density of states g(E), Fermi-Dirac

distribution f(E) and carrier concentration (n for electrons and p for holes) for intrinsic

semiconductors with respect to energy (at room temperature). The expression of intrinsic

carrier concentration for electrons and holes is given by equation 3.18b. If Ei = EF is the

intrinsic Fermi level, the expression for the effective density of states can also be obtained

53


E

g(E)f(E)

Ec

E = EF i

Ev

Carrierdistribution

0 0.5 1.0

E E

Holes (p = n)i

Electron (n = n)i

Valanceband

Conductionband

Eg EF

Figure 3.6.: Schematic representation of band diagram, density of states g(E), Fermi-Diracdistribution f(E) and carrier concentration (n for electrons and p for holes) for intrinsicsemiconductors.

as

Nc = ni exp [(Ec − Ei) /kBT ] (3.20a)

Nv = ni exp [(Ei − Ev) /kBT ] (3.20b)

Again these equations the intrinsic Fermi energy can be expressed as,

Ei =Ec + Ev

2+

3kBT

2ln

(

Nv

Nc

)

=Ec + Ev

2+

3kBT

4ln

(

m∗

h

m∗e

)

(3.21)

equation 3.21 indicates that at T = 0K, the Fermi level of an intrinsic semiconductor

lies in the middle of the band gap (as shown in figure 3.6). For most of the practical

cases m∗

h 6= m∗

e, this non equality in the effective masses for electrons and holes led to a

small deviation of the Fermi level from the middle position. For a constant temperature

the density of electrons and holes in a non-degenerate semiconductor is constant and the

product of the electron and hole carrier concentrations is equal to the square of intrinsic

charge carrier density. This is also referred as law of mass action which can be expressed

as

np = n2i = NcNv exp [(Ev − Ec) /kBT ] = NcNv exp (−Eg/kBT ) (3.22)

The law of mass action indicates that although the electron-hole pairs generate and

recombine continuously, yet the product of electron and hole concentrations remains

constant (averaged in time). It also shows that the product np depends on the effective

conduction band density of states, effective valance band density of states, energy band

54


gap and the temperature. It is very important to mention that this product does not

depend on the individual electron and hole Fermi-level positions. This means that the

regardless of the doping, the product np holds true at a given temperature. One of the

most important parameter i.e. intrinsic carrier concentration can be expressed as

ni = (NvNc)1/2 exp [Eg/2kBT ] (3.23)

Figure 3.7 shows the variation of intrinsic carrier concentration ni with temperature for

8

Germanium

Silicon

Figure 3.7.: Dependence of intrinsic carrier concentration ni on temperature T for Silicon (Si)and Germanium(Ge).

germanium and silicon (equation 3.23). It must be mentioned that for the calculation of

each curve temperature dependence of the energy bandgap has been taken into account

(detailed description of temperature dependence of the energy bandgap is discussed in

section 3.3). From equation 3.23 it is clear that the temperature dependence of the

intrinsic carrier concentration is mainly determined by the exponential factor containing

the energy band gap. The effective densities of the valance and conduction band also

depends on temperature.

3.2.5. Extrinsic semiconductors

One of the advantages of the semiconductors is that their various properties can be

altered by the intentionally adding external impurities. If the impurities are deliberately

55


introduced then the material is called extrinsic semiconductor. The introduced impurity

material is called dopant. In order to provide additional charge carriers, dopants must

be ionized inside the semiconductor. If the external impurities i.e dopant atoms have a

higher vacancy than the atoms of the host material; the ionized donor impurities donate

electrons to conduction band, this leads to excess mobile electrons which can contribute in

the conduction process and these impurities are referred to as donors and the material is

called as n-type semiconductor. The donated electrons participate in conduction process,

while the donor remains positively charged. If the impurities have lower vacancy than

the host material, this lead to incomplete atomic lattice, hence they capture electrons

of the host material i.e holes are generated in valance band. This process leads to a

negatively charged acceptor ions. This type of semiconductor is referred to as p − type

semiconductor.

The energy levels of the donors and acceptors are located within the forbidden energy

gap. These energy levels can be divided in two type of levels depending on their depth

from the nearest band edges. If the donor or acceptor levels are near the band edges,

these impurities are referred to as shallow impurities and the amount of energy require for

a donor to get ionized is close to the thermal energy. On the contrary, deep impurities

require higher energies to get ionized and their energy levels are referred to as deep

energy levels. In most of the cases the deep impurities does not contribute to the free

carriers. They work as recombination centers or trap centers. For shallow impurities,

at room temperature almost entire donor or acceptor sites are ionized.i.e Nd∼= N+

d for

donors and Na∼= N−

a for acceptors. Here, Nd and N+d are the total concentration of

donors and ionized donors respectively,where as Na and N−

a are the total concentration

of acceptors and ionized acceptors. In the case of donors, the electron density n is equal

to the concentration of the donors i.e n = Nd, while in case of acceptors the hole density

p is equal to the concentration of the acceptors i.e n = Na. It emphasizes that the

electrons are majority charge carriers and holes are minority charge carriers for the n-

type semiconductor while for the p-type semiconductors holes are majority and electrons

are minority charge carriers.

Only intrinsic semiconductors are relevant for this work, therefore the discussion is limited

for intrinsic semiconductors. However, extrinsic semiconductors are also very useful and

has been studied in various text books [43].

56


3.2.6. Mobility and drift of charge carriers

The electrons and holes in the conduction and valance band respectively are referred to

as free charge carriers. In the presence of an electric field E, free charge carriers acquire

a drift velocity vd (the electron and hole drift velocities will be in opposite directions),

and provide a net current density J. According to the Ohm’s law

J = σE (3.24)

where σ is the conductivity of the medium. The drift velocity is proportional to the

applied electric filed i.e for electrons and holes we have

vdn = −µeE (3.25a)

vdp = +µhE (3.25b)

respectively. Here the constant µe and µh are carrier mobility of electrons and holes

respectively. Carrier mobility is dependent on the rate of scattering events and is related

to scattering relaxation time τ as µ = eτ/m∗. Now one can write the current density of

the electrons and holes as

Jn = −nevn (3.26a)

Jp = pevp (3.26b)

These current densities are in same direction while the drift velocity for electron and hole

are in opposite directions. From this, the conductivity of the electrons and holes can be

expressed as

σn = neµe (3.27a)

σp = peµh (3.27b)

Hence the bulk conductivity in principle can be expressed as,

σ = neµe + peµh (3.28)

Since typically m∗

e < m∗

h, hence µe > µh, i.e. electrons have a major contribution to the

current. The carrier mobility is a very important parameter in defining the conductivity

of the material. It is determined by the random scattering processes, such as impurity

57


scattering and intrinsic phonon scattering. The impurity scattering includes scattering

from donors (neutral and ionized) and acceptor atoms, which is intentionally added to the

material as well as unintentionally induced impurities and native defects. The intrinsic

scattering includes phonon scattering and carrier-carrier scattering. Out of these, carrier-

carrier scattering has a significant contribution for degenerate semiconductors and under

high field applications. Most of the phonon (lattice) scattering can be divided into two

sections, (i) acoustic phonon (ii) optical phonon scattering. In context of this work and

for most of the practical cases the ionized impurity scattering and phonon scattering

are dominant. Mobility from the interaction of acoustic phonons of the lattice i.e µl

and mobility from the interaction of the ionized impurity concentration i.e µi can be

expressed as

µl ∝1

mc∗5/2T 3/2

(3.29a)

µi ∝T 3/2

NI

(3.29b)

where NI is the total impurity concentration i.e NI = Nd + Na. Then the combined

mobility can be expressed as

µ =

(

1

µl

+1

µi

)−1

(3.30)

It shows that higher doping can lead to a reduced mobility because of ionized impurity

scattering. At first the mobility increases with increase in the temperature because the

carrier spends less time in vicinity of the scattering centers, but with further increase in

temperature lattice vibrations can become dominant, increasing the probability of the

scattering from the lattice and hence decreasing carrier mobility.

So in brief, at a particular value of temperature the mobility has its maximum value

which depends on the impurity concentration.

3.3. Thermal properties of semiconductors

Temperature has a significant effect on the properties of the semiconductors. Due to a

change in temperature, most significant effect arises in the energy band structure and

the carrier concentrations of semiconductor materials.

58


3.3.1. Thermal expansion coefficient

The fundamental change that a variation in temperature brings in a semiconductor is a

change in its inter atomic spacing i.e. lattice parameters. For most common semiconduc-

tors the inter atomic spacing increases with increase in temperature. Due to this increase

in the inter atomic spacing material expands. The parameter which usually determines

the expansion of the material is called thermal expansion coefficient αth. The thermal

expansion coefficient also increases with an increase in temperature up to the melting

point of the semiconductor material. Figure 3.8 shows the variation of the lattice param-

Exp

an

sio

nco

effic

ien

t

Figure 3.8.: (a)Variation of lattice constant a as function of temperature for germanium. (b)Temperature dependence of thermal expansion coefficient αth for germanium [44].

eter and thermal expansion coefficient of germanium with temperature. As this work is

related to the melting and solidifying of germanium, it is important to point out that

germanium and silicon expands during the solidification process [45] [46].

3.3.2. Effect of temperature on energy bandgap

The band structure of a semiconductor material is a function of crystal lattice spacing.

Increase in the temperature can cause lattice dilation which contribute to the change in

energy positions of the band and band structure itself. The second contribution comes

from electron-phonon interactions, which also increases with temperature. For most

common semiconductors, the temperature coefficient of the position of band states lie

within the range 10−4 to 10−3 eV K−1. The dilation effect contributes about 20-50 percent

of the total coefficient. The variation of band gap with temperature for a semiconductor

59


can be expressed by the following universal relation,

Eg(T ) = Eg(0)−αT 2

T + β(3.31)

where Eg(0)(eV), α(eV/K) and β(K) are constants for a particular material and α and β

> 0. The temperature coefficient dEg/dT is negative for silicon (Si)and germanium (Ge)

but some semiconductors also have positive temperature coefficient such as PbS. Figure

Germanium

Silicon

Material E (eV)g α (eV/K) β(K) E (eV)g

Ge 0.747 4.77 x 10-4 235 0.66 (300K)

Si 1.125 4.73 x 10-4 636 1.12 (300K)

Figure 3.9.: Temperature dependence of minimum energy band gap (Eg) for silicon and germa-nium. Table in the inset of the plot shows the values of coefficients for silicon and germaniumused to evaluate the curves from equation 3.31.

3.9 shows the variation of band gap Eg with temperature T for silicon and germanium

using equation 3.31. The value of bandgap decreases with increase in the temperature.

We also know that the bulk conductivity can be expressed by equation 3.28. For intrinsic

semiconductors, the expression of n and p can be written as (using equation 3.23 and

effective density of states for electrons and holes)

n = p = constant× T 3/2 exp [−Eg/2kBT ] (3.32)

hence, the conductivity can be expressed as

σ = constant× e (µe + µh)T3/2 exp [−Eg/2kBT ] = σ0 exp [−Eg/2kBT ] (3.33)

The constant σ0 = constant×e (µe + µh)T3/2 and it includes the hole and electron mobil-

ities. These mobilities depend on temperature as T−3/2, but this function is canceled out

by the temperature dependence term T 3/2. Thus, the slope of the curve ln σ with respect

60


to 1/T , i.e. (−Eg/2kB), provides the value of energy band gap of the semiconductor.

3.3.3. Effect of temperature on carrier concentrations

In order to understand the behavior of the semiconductor devices with respect to tem-

perature, one need to understand the dependence of the carrier concentrations on tem-

perature in addition to the dependence of Ef . In case of an intrinsic semiconductor, the

temperature dependence of the intrinsic carrier concentration is discussed in the section

3.2.4. For an extrinsic semiconductor, the behavior of carrier concentration with respect

to temperature can be understood by plotting n/Nd as a function of temperature. There

100 200 300 400 500 6000.0

0.4

0.8

1.2

1.6

2.0

n/N

d

T(K)

Intrinsic regime

Freeze-out regime

Saturation regime

Figure 3.10.: Temperature dependence of majority carrier concentration in a dopedsemiconductor.(n-type silicon with donor concentration Nd = 1015/cm3)

are three distinct regions in figure 3.10. At low temperatures, carriers are generated by

the ionization of dopants. This region is referred to as Freeze-out region or Freeze-out

regime. As the temperature increase further and reaches the intermediate range, the con-

tribution from donors decreases due to the ionization of all available donor sites. Apart

for that, in this intermediate range the temperature is also not sufficient enough to ther-

mally excite the intrinsic charge carriers. Hence in this range, the carrier concentration

is constant with respect to temperature and this region is referred to as saturation region

or saturation regime.

At higher temperatures, carrier concentration in conduction band is provided by the

thermal excitation of electrons from the valance band to the conduction band. This

region is referred to as intrinsic region or intrinsic regime. In intrinsic regime, the extrinsic

61


semiconductor behaves as an intrinsic semiconductor. The saturation region is the most

important regime for semiconductors.

With the increase in temperature, mobility of the electrons and holes decreases. It is

due to the reduction in the life time of charge carriers. The temperature dependence of

electron mobility, hole mobility and drift(ohmic) mobility of electrons has been explained

by various researches [47]. Furthermore, one can also determine the variation of conduc-

tivity with respect to the temperature. The conductivity of an intrinsic semiconductor

is shown in equation 3.33.

1/T (K )-1

Intr

insic

co

nd

uctivity

cm

Ω-1

-1

Figure 3.11.: Intrinsic conductivity σ of germanium as a function of reciprocal of temperature.

Figure 3.11 shows the variation of conductivity with respect to temperature for an intrin-

sic semiconductor (germanium). It is clear from the plot that as temperature increases,

the conductivity also increases. This is in agreement with the fact that increase in tem-

perature also increases the carrier density for the intrinsic semiconductor. This argument

of increase in conductivity is also true in case of extrinsic semiconductor at high tempe-

ratures.

For an extrinsic semiconductor, one can express conductivity for a n-type semiconductor

at low temperatures as

σ = σ0exp [− (Ec − Ed) /2kBT ] (3.34)

In this case, the plot ln σ Vs 1/T also gives a straight line with slope (Ec − Ed) /2kB.

62

3.4. Optical properties of semiconductors


The interaction of electrons and photons in a semiconductor is very essential in the con-

text of this work. There are several mechanisms which can contribute to the optical

properties of semiconductors, e.g. optical absorption, recombination of charge carriers

and photoconductivity. Most useful representation for the optical properties of a semi-

conductor is its complex dielectric function.

3.4.1. Optical absorption in semiconductors

There are several optical processes which affect the total optical absorption in semicon-

ductors. All these mechanisms contribute towards the total optical absorption coefficient

of the material, e.g. αab. These mechanisms can be defined as

• fundamental absorption

• exciton absorption

• absorption due to dopants, imperfection and impurity

• absorption due to interband transitions

• free carrier absorption.

When photon energies are greater than the energy band gap, the most significant con-

tribution comes from the electronic transitions between the valance band states to the

conduction band states. For energies below the energy band gap, absorption is mostly due

to excitons and transitions between the impurity and interband states. These transitions

occur between the acceptor energy levels to the conduction band and donor energy levels

to the valance band. Other than this, free carrier absorption arises due to the transitions

to high energy level within the same energy band. During an electronic transition, the

energy E and momentum ~k must be conserved. In this section, a review for some of the

basic mechanisms of optical absorption in an intrinsic semiconductor is presented using

germanium as an example. Figure 3.12 shows the optical absorption spectra obtained

by Philip and Taft [48]. The spectrum was obtained for a single crystal of germanium

at 300K using reflectivity measurements. Figure 3.13(a) shows the optical absorption

spectra by Newman and Dash for 300K and 77K. It was obtained for a single crystalline

germanium sample in the range of 0.5 eV (620nm) to 2.0eV (2480nm) [49]. Figure 3.13(b)

63


Figure 3.12.: Spectral dependence of optical absorption of germanium for a range from 1eV to10eV as reported by Philip and Taft at 300K for single crystal material [48].

0.80eV(Direct bandgap)

0.66eV(Indirectband gap)

Figure 3.13.: (a) Absorption spectra for the single crystal germanium for low energy range of0.5eV to 2eV as reported by Newman and Dash (1955) at 300K and 77K [49]. (b) Expandedview of the left curve.

64


shows the extended view of the figure 3.13(a) in horizontal scale. There are several mech-

anisms which contribute towards the total optical absorption in a medium. The most

important mechanisms are briefly discussed in this section. These mechanisms also help

to understand the distinct features of the optical absorption spectra (shown in figure

3.13). The high energy features of the spectra of figure 3.12 are due to the transitions

occurring in the preferred directions of the crystal lattice planes.

3.4.1.1. Fundamental absorption process: intraband electronic excitations

In this process, a photon excites (provides sufficient energy) an electron from the valance

band to the conduction band. The condition for the conservation of momentum leads to

two types of interband transitions,

• if the minima of the conduction band and the maxima of the valance band does not

lie at same value of the wavevector k, the transition is called “direct transition”.

• if the maxima of the conduction band and minina of valance band does not occur

at the same value of wavevector k, the transition is called “indirect transition”. For

indirect transitions, the conservation of the momentum requires the involvement

of an extra particle called “phonon”. The fundamental absorption is weaker for

indirect-gap semiconductors in comparison to direct-gap semiconductors.

By using the expression for the transition probabilities and density of states, the pro-

portional expression for the absorption coefficient in case of direct transitions can be

obtained as

αab = Aab (hν − Eg)γ (3.35)

here, Aab is a constant, hν refers to the photon energy (in eV), Eg is the band gap (in

eV), γ is a constant giving information about allowed and forbidden transitions and αab

is the absorption coefficient in cm−1. In case of germanium, the direct band gap occurs

at 0.805eV (300K). Figure 3.13 shows a very abrupt decrease (with decrease in photon

energy) in the absorption coefficient at ≈ 0.805eV for 300K (the position is marked by

a red line in figure 3.13(b)), this “knee” like feature is attributed to the direct electronic

band gap.

Furthermore, moving towards the low energy shoulder of this curve decreases with higher

slope at around 0.66eV in the absorption coefficient of germanium. The indirect energy

band gap of germanium is at 0.66eV (the position is marked by a blue line in figure

65


3.13(b)), which means that this second decrease in absorption coefficient is due to the

indirect transitions in germanium. For indirect transitions three particles (i.e photon,

electron and phonon) are involved in the process, to satisfy the conservation of energy and

momentum. In contrast to a photon, a phonon has lower energy but higher momentum.

Thus, for an indirect transition the momentum is conserved by the absorption or emission

of a phonon with energy Ep. The expression of αab in case direct transition can be

modified to include the absorption and emission of a phonon and can be expressed as

αaba =Aaba (hν − Eg + Ep)

γ

exp (−Ep/kBT )− 1; absorption of phonon (3.36a)

αabe =Aabe (hν − Eg − Ep)

γ

1− exp (−Ep/kBT ); emission of phonon (3.36b)

respectively. Here, the allowed and forbidden transition values of γ are 2 and 3, respec-

tively. The denominator term in equation 3.36 provides the number of phonons according

to Bose-Einstein statistics. For hν > Eg + Ep, the probability of both these processes

occurring are equal. So, the total absorption coefficient can be expressed as

αab = αabe + αaba (3.37)

Due to the involvement of a phonon, the probability of the absorption of light in a indirect

transition is much lower in comparison to direct transition. At the absorption edge, the

absorption coefficient varies from 104 to 105 for direct transition and 10 to 103 for indirect

transitions.

3.4.1.2. Absorption due to excitons

An exciton (electron-hole bound pair) is typically formed at low (cryogenic) temperature

due to the Coulomb interaction between an electron-hole pair. The excitonic states lies

just below the conduction band and can exist in a series of bound states with in the

energy gap. This means that absorption process due to excitons is observed at energies,

which are slightly lower than the energy-gap of a semiconductor. Excitonic absorption

is typically observed at temperatures where kBT is smaller than the excitonic binding

energy. These excitons can move through the crystal lattice without transporting a net

charge. At room temperature the kBT is ≈ 0.029eV while the binding energy of a exciton

in a semiconductor is ≈ 0.01eV which reduces the possibility of absorption due to exciton.

66


3.4.1.3. Absorption due to dopant, imperfection and impurity

These absorption processes are visible in the low energy shoulder of the absorption edge.

Transitions between the impurity state and the band state (acceptor or donor level to

conduction band or valance band) are responsible for this type of absorption. These

transitions determine the absorption in far-IR region. This type of absorption is also

enhanced by the imperfection in crystal lattice which introduces localized energy levels

in the energy gap. These imperfections include point defects, unintentional impurities,

grain boundaries and dislocations. Contributions to the absorption coefficient also come

from transitions occurring from localized levels to the conduction band or valance band

and also from transitions within these imperfection levels.

3.4.1.4. Absorption due to intraband transitions and free carriers

The transitions occurring within the conduction or valance band also contribute to the

absorption coefficient. Within the valance band, these transitions arise due to the pre-

sence of light-hole (LH), heavy hole (HH) bands or split-off bands. The absorption of

photons can occur between these LH, HH and split-off bands and also within LH and

HH bands itself.

Free carrier absorption is due to the transition of an electron to a higher energy level

with in the same energy band. These transitions contribute towards the absorption

continuum at lower energies. This process requires the participation of a phonon and can

be understood similar to an indirect transition process. The absorption coefficient due

to free carrier absorption αabfc is proportional to the λp (where p lies between 2 and 3)

and inversely proportional to carrier mobility µe. If the mobility is large, the scattering

is weak and the absorption coefficient becomes smaller.

3.4.1.5. Temperature dependence of optical absorption

As discussed in section 3.3, a change in temperature results in a change of the energy

bandgap and the carrier concentration of semiconductors. For most common semicon-

ductors (such as germanium and silicon), the energy gap decreases with the increase

in temperature. It means that the long wavelength (low photon energy) edge of the

absorption coefficient is red shifted.

67


It was suggested by Newman and Dash [49] that in the long wavelength range, some

sections of the two curves (for 300K and 77K) shown in 3.13 can be superimposed using

a horizontal shift accompanied by a small vertical shift. The shift required to overlap the

curves for these two temperatures ranges from 0.07eV to 0.12eV. Just below the direct

transition ’knee’ feature only horizontal shift of 0.07eV is required [49], which means

that the direct band gap region of the absorption spectra shifts to lower energies almost

linearly with temperature.

An increase in temperature not only decreases the indirect band gap edge of the absorp-

tion coefficient, but also alter the phonon-electron interactions. While the decrease in

the band gap shifts the indirect band edge to red, the increase in phonon interactions

increases the absorption coefficient. Hence, to superimpose the curves we also need to

accommodate the vertical shift. According to Newman and Dash [49], the best super-

imposition can be achieved by a horizontal shift of 0.01eV and then a vertical shift by

multiplying with a factor 2.

3.4.1.6. Pressure dependence of optical absorption

Like temperature, the change in pressure also results in the change of the energy band

gap of a semiconductor. For most semiconductors, the energy band gap decreases with

increase in pressure. But there are some materials such as germanium for which the

energy band gap first increases with pressure and reaches a certain maximum value.

After this optimum value of pressure, any further increase in pressure decreases the

energy band gap. Slykhouse and Drickamer studied this behavior for germanium [50].

For germanium, the inverse pressure dependence of the band gap arises due to different

pressure dependencies of the two conduction band minima. The (111) conduction band

minimum rises and the (100) conduction band minimum drops with increasing pressure.

The reversal of pressure coefficient occurs when indirect transitions in the (100) direction

can occur at lower energies than those in the (111)direction. The initial blue shift of the

energy band gap can be approximated as ≈ 8.0×10−6 eV/bar. For pressures higher than

50,500bar, its sign changes and it has a red shift with increasing pressure. This change

in the energy band gap also affects the low energy optical absorption edge depending on

the sign of pressure coefficient of the energy band gap. [50]

68


3.4.1.7. Reflection spectra

One of the important tools to investigate the optical absorption is the measurement of

the reflectance of the sample. If I0 is the incident light intensity, I is transmitted light

intensity, R is the reflectivity of the material, then the transmission T = I/I0 can be

expressed as (neglecting interference)

Tr =(1−R)2 exp (−αabd)

1−R2 exp (−2αabd)(3.38)

where d is the material thickness. For large αabd ≫ 1

Tr = (1−R)2 exp (−αabd) (3.39)

If we ignore the reflection then the transmitted intensity can be expressed as

I = I0 exp (−αabd) (3.40)

which is the as Lambert-Beer law. Tr and R are the two important experimentally

measurable quantities. The analysis of T or R with respect to λ or the energy of photons

helps us to obtain the real and imaginary part of the dielectric function. Figure 3.14

Figure 3.14.: Optical reflectance spectra for germanium single crystal at 300K [51]

.

shows the reflectance spectra for single crystal germanium at 300K [51].

69


3.4.2. Recombination process

Recombination of carriers (electron and hole) in a semiconductor is the process in which

an electron and a hole are recombined and energy is released as radiation or in terms of

heat. If optical absorption is responsible for the generation of carriers due to excitation of

electrons, recombination restores this generation excess carriers. A recombination process

can be described as radiative or non-radiative depending on whether the recombination

process leads to emission of a photon or not. Similar to the absorption process, in

direct semiconductors the most likely transition for recombination is across the minimum

energy gap, i.e. between a occupied state at the minimum of the conduction band and

an unoccupied state at the maximum of the valance band. These transitions are most

likely responsible for radiative recombination processes. In the indirect semiconductors

(at the minima of the band gap), the recombination process requires an extra particle

i.e. phonon. This reduces the probability of indirect process in comparison to the direct

transition process.

The emission spectra is (i) intrinsic, when the process is across the fundamental or edge

of the gap and (ii) extrinsic (characteristic or activated emission) when it is assisted by

impurities.

3.4.2.1. Radiative recombination process

There are several mechanisms which lead to the emission of photons during a radia-

tive transition. A schematic diagram involving several possibilities of transition for the

radiative recombination is shown in figure 3.15.

Process 1: Intraband transition within the conduction band. In this process an electron

excited well above the minimum of the conduction band got de-excited and reaches the

minimum of the conduction band attaining the thermal equilibrium with the lattice.

Process 1 is a phonon assisted process and leads to the emission of a phonon assisted

photon or only a phonon.

process 2: Interband transition. In this process the direct recombination between an

electron and a hole across the energy gap result into an emission of a photon of energy

Eg = hν. This is most likely intrinsic process. Although this process occurs near the

band edges, the thermal distribution of carriers can lead to a broad spectrum.

Process 3: Decay of an exciton which leads to the emission of a photon. This process

70


Ec

Ev

Ee Ed Ed

EaEa

1

2 3 45

6 7

excitedstates

Eg

Figure 3.15.: Schematic diagram showing various radiative combination processes. (1) Intra-band transition within the conduction band, (2)Interband transition, (3) Decay of an excitonwhich leads to the emission of a photon, (4) Donor to free-hole recombination, (5) Free elec-tron to acceptor transition, (6) Donor to acceptor pair transition (7) Excitation and radiativede-excitation of electrons from an impurity state which has completely filled inner shells.

is most likely to occur at very low temperatures due to lower binding energy of exciton,

e.g. cryogenic temperatures. These transitions are not essentially important in context

of the present work as at room temperatures or above the probability of formation of an

exciton is neglected.

Process 4 to 6 are transitions which start and/or finish on localized states and band

edges. Process 4 represents donor to free-hole recombination process (transition) which

is labeled as D0h, process 5 represents a free electron to acceptor transition labeled as eA0

and process 6 is donor to acceptor pair transition. These transitions play an important

role for achieving a required luminescent spectrum. The energy of such a transition is

small compare to a band to band transition. In impurity states, the electron is highly

localized in space and its wavefunction extends to its nearest neighbors. The energy of

a deep level extends in to a wide range of k values. Hence, direct transitions from an

impurity level to a wide range of extended states are allowed without the assistance of a

phonon. This is very relevant in case of indirect-gap semiconductors e.g. germanium.

Process 7, represents the excitation and radiative de-excitation of electrons from an im-

purity state which has completely filled inner shells, e.g. impurity of rare earth elements

or transition metals.

71


3.4.2.2. Nonradiative recombination process

Other than the radiative recombination processes, recombination process can also occur

via a nonradiative mechanism. This process can include

• multiple phonon emission, where the energy is converted in terms of heat and

dissipated.

• Auger effect, where the energy of a transition is absorbed by another electron which

rises to a higher energy state in conduction band (in a special case a second electron

can also get sufficient energy to leave the material and the energy of the second

excited electron can be dissipated through heat.

• recombination due to surface states and defects

Elastic strain can introduce shallow levels. A variety of points defects can also introduce

the energy levels in the band gap contributing towards the recombination process. Energy

levels arising due to other defects such as vacancies, interstitials, dislocations, stacking

faults, grain boundaries are usually located deeper in energy gap and have localized levels.

In general extended defects as dislocation and grain boundaries are detrimental in device

applications.

3.4.2.3. Recombination rates

For a transition occurring across a direct gap, the recombination rate is proportional to

the electron and hole concentrations

Re = Recnp (3.41)

here Rec is the recombination coefficient, which is related to thermal generation rate Gth

by Rec = Gth/n2i . At thermal equilibrium, we have np = n2

i , which means that Re =

Gth and the net transition rate U (= Re −Gth) is zero. By studying the recombination

process, one can also obtain the life times of the carriers. For a n − type material the

carrier life time for the holes can be expressed as

τp =1

RecND

(3.42)

72


similarly for p− type material the carrier life time for the electrons is

τn =1

RecNA

(3.43)

For indirect-gap semiconductors, the dominant transitions are indirect recombinations

which occur through the trap levels in the energy gap. If Nt is the volume density of the

trap levels in the energy gap. The process of the recombination through the band gap

is assisted by the defect levels (trap levels) and described by Shockley-Read-Hall(SRH)

recombination model [52]. Using this model, the net transition rate can be expressed by

U =Ntσncσpcvth (pn− n2

i )

σnc

[

n+ ni exp(

Et−Ei

kBT

)]

σnp

[

p+ ni exp(

Ei−Et

kBT

)] (3.44)

here, σnc and σpc are electron and hole capture cross-sections, respectively. Et is the

energy level of the trap in the bandgap, Ei is the intrinsic Fermi level and vth is the carrier

thermal velocity. Here the term np− n2i determines the generation or the recombination

of the carriers. From the law of mass action at the thermal equilibrium where np = n2i ,

one can obtain that U = 0 (using equation 3.44).

Another important observation of equation 3.44 occurs when Et = Ei, it is the condition

for maximum U . It indicates that the levels present at the mid-gap are the effective

recombination or generation centers. For this process, the life time of holes in a n-type

material can be expressed

τSRHp =

1

σpcvthNt

(3.45)

similarly for p− type material the carrier life time for the electrons is

τSRHn =

1

σncvthNt

(3.46)

The life time arising from the indirect transition is inversely proportional to the trap

density and for direct transition (equation 3.42 and 3.43) it is inversely proportional to

the carrier concentration or doping concentration. If both the radiative and nonradiative

centers are present inside the semiconductor, then the effective carrier life time is

τ =

(

1

τr+

1

τnr

)

=τrτnrτr + τnr

(3.47)

There are several nonradiative recombination processes involved and one has to sum over

73


all the nonradiative processes. The radiative recombination efficiency is defined as the

ratio of the radiative recombination rate and the total recombination rate. Thus, the

efficiency can expressed as

ηr =1

1 + (τ/τnr)(3.48)

In general, η is a function of temperature, concentration of dopants and presence of

various defects and impurities. In most of the cases nonradiative transition cross-sections

are higher than the radiative transition cross-sections.

It is important to mention that for bulk semiconductors, apart from the radiative and in-

ternal nonradiative recombinations we must also include the nonradiative surface recom-

binations. Then, the total measured life time for a bulk semiconductor can be expressed

as1

τbulk=

1

τr+

1

τnr+

1

τsurf(3.49)

The τsurf is determined by the recombination centers at the surface. These surface states

arise due to the abrupt change in the surface (metal-semiconductor contact), impurity or

oxide layers at the surface, this can introduce discrete energy levels in the band gap. It

is important to point out the difference between a trapping center and a recombination

center. An imperfection site is referred to as trap center, if the captured carrier at the

site has a greater probability of being thermally re-excited to a free state in comparison

to recombining with another carrier of opposite sign. For a recombination center the

captured carrier has a greater probability of recombining with a carrier of opposite sign

at the imperfection site rather than of being re-excited to a free state.

3.4.3. Photoconductivity in semiconductors: Intrinsic

photoconductivity

The absorption of a photon can lead to the generation of free charge carriers in form of

a hole and an electron. This basic process of the generation of free charge carriers in

a semiconductor by means of optical excitation provides the basis of photoconductivity.

For a very simple case of an intrinsic semiconductors, optical excitation leads to an

electron in the conduction band where it can contribute to an increase of conductivity.

At the same time a hole is generated in the valance band which further enhances the

conductivity. If the semiconductor is impure, then the electron from the impurity level

can rise to the conduction band or a hole is generated by exciting an electron from the

74


valance band to the acceptor levels. The generation of electrons and holes can only occur

when the exciting radiation has sufficient energy, i.e. the energy of the incident photon

hν is greater than the energy of the involved gap.

After the excitation of carriers, photoconductivity can be observed until the generated

carriers recombine, i.e. the life times of excited carriers. The motion of these carriers

under the influence of an external field (e.g. electric field) also plays a major role in the

occurrence of photoconductivity.

Photoconductivity is mostly measured in form of generated photocurrent. Hence the ba-

sic process and magnitude of photo current involves the generation of electrons and holes

due to optical absorption, transport of these electrons and holes through the material

under the influence of the electric field and recombination of the photo-excited electrons

and holes. Analysis of the measured photocurrent in experiments is influenced by any of

these three stages independently. In a simple case, monochromatic light of appropriate

wavelength is incident on the material surface. This will result in to the generation of

equal densities of electrons and holes, i.e. δn = δp leading to a change in the conductivity

of the material as,

δσ = σin = e (µnδn+ µpδp) (3.50)

where σph is the contribution of conductivity due to photo-generated charge carriers. One

of the important parameters determining the photoconductor is the responsivity, which

is defined as

S = Iph/Pin (3.51)

where Iin is the photon generated current and Pin is the incident photon power on the

sample. Figure 3.16 shows the schematic diagram for photoconductivity. Vsample is the

volume of the sample and Aelec is the area of the electrode. The distance between the

electrodes is l and the sample thickness along the direction of light is d. A fraction

of incident radiation is absorbed in the sample and generate charge carriers, which can

generate a photocurrent if external voltage is applied on the sample.

The quantum efficiency η of the material is defined as the fraction of the photons that

produce the charge carriers. Under the following assumptions,

• that the energy of one photon, hν is low enough that a single photon can produce

at most one conduction electron

• that every absorbed photon generate a conduction electron

75


e eh

h

h e

h

e

Aelec

l

Incident light

VSAMPLE

+-V

d

Pph

Figure 3.16.: Schematic diagram showing the process of photoconductivity

The quantum efficiency η can be defined as

η = 1− eαabsd (3.52)

where, d is the thickness of the sample along the incident direction of photon and αabs

is optical absorption coefficient. It is important to note that if the product of αabs d is

large enough such that eαabsd ≪ 1 than η is unity, which can be achieved by increasing

the thickness of the sample. The power absorbed inside the sample can be written as

Pinη.

The total number of photons absorbed per second can be expressed in terms of incident

power as Pinη/hν. If the average life time of the electron before recombination is τ ,

then the average density of electrons and holes arising due to photoconductivity can be

expressed as

nph = pph =Pinητ

hνVsample

(3.53)

Now these electrons can be accelerated by the external electric field E due to applied

voltage V . The mobility of an electron is affected by a variety of scattering mechanisms

and its interaction with phonons. Due to the effect of an electric field the electrons

acquire a means or average velocity. Assuming that the electric field and current are

76


parallel to each other, one can write the photocurrent as,

Iph = enphAelec〈vn〉+ epphAelec〈vp〉 (3.54)

If the electron and hole average velocity is included in terms of their mobilities, then

expression for photocurrent in terms of incident power can be expressed as

Iph = Pphe

VsamplehνAelecητ(µn + µp)E (3.55)

Using this equation the responsivity can be expressed as

S =e

hνηG (3.56)

where Gph is the photo conductive gain There are several physical and structural pro-

perties which affect the generated photocurrent. One of them is the presence of metal

barriers, which arise due to the contacting of a photoconductor with an external circuit.

The presence of the these barriers prevents the free flow of electrons and holes. The

presence of a metal contact with the photoconductor can affect the photocurrent due to

• improper matching of work functions between the metal and the semiconductor.

• presence of surface states on the semiconductor, producing an intrinsic surface

barrier

• presence of a thin layer of a third material (such as an oxide) which in turn causes

barriers for reason due improper matching or surface states.

The presence of impurities also affect the photoconductivty of the material, as these

impurities can create trap centers and decreases the photocurrent.

Another parameter which affect the photoconductivity is the presence of crystal defects

in lattice. These defects can be formed by the thermal dissociation or disorder. When

the material is heated to a high temperature or exposed to an excess pressure these

impurities get incorporated, leading to a defect state or disorder in the crystal lattice.

77

Chapter 4Basic theory for filling

In the context of this work, the filling properties of different materials (in liquid phase)

inside the hole of a capillary are very important. Metals and semiconductors in molten

phase (e.g. gold and germanium in liquid phase) show a different behavior in comparison

to the common liquids, e.g. water, when filling into the capillary.

The surface tension of a liquid is the measurement of the cohesive energy present at any

given interface. Inside a liquid, each molecule is pulled equally in every direction by

neighboring liquid molecules, resulting in a net force of zero on molecules. The molecules

at the surface are not surrounded by the liquid molecules and are therefore are pulled

inward. This creates some internal pressure (or force) and compel the liquid surface to

contract to the minimal surface area. At the same time, the desire to minimize its energy

state forces the liquid to reduce the number of surface molecules by reducing the surface

area.1 The enhancement of the intermolecular attractive forces at the surface is called

surface tension and has units of force per unit length.

A liquid with higher surface tension in contact with a solid does not allow its surface

molecules to have a strong bond with the molecules of the solid, due to their higher

intermolecular attraction with in the liquid.

The surface tension of a pure germanium melt has been studied by various researchers

[53] [54]. The reported experimental values by these researchers show slight discrepancy,

which has been attributed to the presence of impurities (e.g. oxygen), which can cause

a change in the measured value. In this work, the value of surface tension of germanium

1This the reason why the liquid drop attain a circular geometry.

79

4. Basic theory for filling

is taken as ≈ 600mN/m [53].

4.1. Contact angle and curvature of the meniscus of

liquid

The behavior of the surface of liquid in contact with a solid surface depends on the nature

of interactions exist between the molecules of the liquid, vapor and solid surfaces. The

curvature of the surface of a liquid, which is in contact with both the atmosphere and the

solid, is called the meniscus. The angle of the tangent of the curvature to the surface of

the solid is called the contact angle, denoted as θ. The contact angle of a liquid is specific

for any given system and is determined by the interactions (surface tension) across the

interfaces between liquid, solid and surrounding vapor (atmosphere). Figure 4.1 shows

(a) (b)

Liquid Liquid

θ

θr

R

r

R

Glass capillary

Figure 4.1.: Schematic representation of the curvature of two different types of liquids insidethe glass capillary. (a) Concave curvature (b) Convex curvature.

the two possible situations for the curvature and contact angle of the liquid surface with

respect to the inner surface of the capillary.

• If the molecular forces within the liquid are weaker in comparison to the forces

between the solid and the liquid (e.g. water in a silica capillary), then molecules

are pulled towards the surface of the solid and a concave curvature is obtained.

80

4.2. Capillary effect

This leads to a contact angle between 0° and 90° as shown in figure 4.1(a). These

liquids are called wetting liquids, e.g. water .

• If the molecules of a liquid show stronger attraction to each other than to the

molecules of a solid (capillary), then liquid molecules are not attracted towards the

surface of the solid, as a result the contact angle lies between 90° and 180°. These

types of liquids are called anti-wetting liquids, e.g germanium or gold in liquid

state.

The contact angle of a liquid in a specific environment depends of the surface tensions

of liquid/vapor (γlv), solid/vapor (γsv) and solid/liquid (γsl) interfaces.

cos θ =γsv − γslγlv

(4.1)

It is clear from the equation that if the γsl has a higher value then γsv, the cos θ is negative

leading to a contact angle of more then 90°, which is the case for germanium and gold.

4.2. Capillary effect

One of the important effects related to the contact angle and the surface tension of the

liquid is the capillary effect. If one end of the capillary is inserted into the liquid in

a vertical position, then the nature of interaction between the molecular forces of the

liquid and the solid can cause the liquid to move vertically upwards or to be pushed out

of the capillary. This phenomena is called the capillary effect and observed for liquids

which have concave curvature with a contact angle between 0° and 90°. The cause of the

capillary effect is that the molecules wetting the inner surface of a tube move upwards

and the surface tension of the liquid induces a net force on the remaining molecules,

which pulls the liquid upward as well. The column of the liquid stops when the upper

end of the capillary is reached, or when the capillary force is at an equilibrium with the

force due to the gravity of the liquid column.

The reverse of capillary action occurs for the liquids with convex curvature, the liquid

molecules show a stronger attraction with each other, which creates a net reverse force,

pushing the liquid out of the capillary. The critical pressure required to overcome this

reverse action of capillary effect can be obtained using Young−Laplace equation, which

describes the capillary pressure difference across the interface between two static fluids

81


(air and liquid, in this case) due to the phenomenon of surface tension. It relates the

pressure difference to the shape of the surface. Using this equation, the pressure created

by the surface tension of the liquid to force the liquid in or out of the capillary can be

expressed as

Pc =2γ

rcosθ (4.2)

γ is the surface tension of the liquid, r is the inner radius of the capillary. Equation

4.2 indicates that for a liquid showing a reverse of the capillary effect ( 90°< θ < 180°),

a critical pressure Pc must be applied to fill the liquid into the capillary. The Inverse

proportionality of the radius also indicates that it is difficult to fill these materials in

smaller holes. The contact angle for fused silica and pure germanium melt is found to

be in range of 150°, [53], 2 indicating that germanium shows an anti-capillary behavior

with silica. For a capillary with a hole radius of 1µm, the value of critical pressure is ≈11bar.

Other then the surface tension and contact angle, viscosity also affects the flow of a

liquid inside a capillary. The viscosity of liquid describes its internal property to oppose

or resist the flow of liquid and can be thought of as a measure of fluid friction. The

viscosity is a function of temperature and tends to decrease (or, alternatively, its fluidity

tends to increase) as the temperature increases. For germanium, it varies from 0.13Pa.s

to 0.70Pa.s over a temperature range of 1160K to 1425K [55]. In this work, the viscosity

of germanium melt is taken as an average between these values, i.e. ≈ 0.40Pa.s.

4.3. Washburn’s equation

The theory of the penetration of liquids into cylindrical capillaries with a small inner

diameter was published in detail by Edward W. Washburn in 1921 [56]. He concludes

that the surface tension, viscosity and contact angle play major roles in determining

the flow of the liquid into a capillary. Washburn’s work is modified here to match the

requirements for filling a Ge- or Au- melt. The Hagen-Poiseuille law describes the laminar

flow of a viscous and incompressible liquid through a cross-section with a circular shape

and a constant diameter. According to this law the rate of the flow of the liquid can be

2It has been also reported that this wetting angle reduces exponentially to 117° with time over a timeperiod of 100hrs, but in context to this work it is assumed to constant as the time duration rangesfor 5-10 minutes [53].

82

4.3. Washburn’s equation

written as,dl

dt=

Ptot

8r2ηl

(

r4 + 4csr3)

(4.3)

here, dl is the corresponds to the length of the volume dV = πr2dl which liquid covers in

time dt, Ptot is the sum of all the pressures acting on the liquid, r is the inner radius of

the capillary, η is the viscosity of the liquid, cs is the coefficient of slip and l is the total

length of the liquid column. The slip coefficient defines the behavior of the liquid with

the solid boundary of the capillary. The value of the cs is zero for wetting material.

The total pressure includes the atmosphere pressure Pa, pressure due to capillary force

Pc = 2γ cos θ/r and hydrostatic pressure (Ph = lgρ sinφ). The total pressure can be

expressed as

Ptot = Pa + lgρ sinφ+2γ

rcos θ (4.4)

The pressure Pa is equal to the pressure difference between the two ends of capillary,

which is equivalent to the additional pressure applied from one end of the capillary. The

capillary pressure is due to the capillary effect, which, for a wetting material is positive

and supports the capillary filling. ρ is the density of the liquid at the filling temperature.

φ is the angle of the capillary with the horizontal plane and g is the gravity acceleration.

If the filling capillary is parallel to the horizontal plane, then hydrostatic pressure can be

assumed to be zero. Also for capillaries with small diameters, the gravity effect can be

neglected (in context of this work). In this case 4.3 can be written as,

dl

dt=r2Pa + 2γr cos θ

8ηl(4.5)

The variables r, θ, ρ, η and Pa are assumed to be independent of time and length and

are therefore treated as constants. Integrating equation 4.5 over time to obtain the total

filling length gives,

l =

√

r2Pat

4η+trγcosθ

2η(4.6)

Equation 4.6 can be plotted for different diameters in order to estimate the filling length

at a given pressure for a germanium-melt. Figure 4.2 shows the variation of filling length

with pressure for different diameters of the capillary for a germanium-melt. It is clear

from the figure that a critical pressure must be applied to start the process of filling.

Also, as the hole diameter decreases, the value of the critical pressure increases. Once the

applied pressure exceeds the critical pressure, the rate of filling is essentially dependent

83


0 50 100 150 2000

20

40

60

80 1000nm

800nm

600nm

400nm

200nm

Fill

ing

le

ng

th (

cm

)

Pressure (bar)

Figure 4.2.: Variation of filling length with applied pressure for different diameters forgermanium-melt. These curves are obtained by using equation 4.6.

on the pressure and viscosity of the liquid.

The parameter of critical pressure is the most important parameter when designing a

setup for filling germanium or gold.

84

Chapter 5Material properties

5.1. Dielectric: silica

All fibers and capillaries used in this work are made of dielectric material fused silica, i.e.

SiO2, which is an amorphous form of silica. Synthetically fabricated silica tubes and rods

of various purity levels were obtained from commercial manufacturer “Heraeus”. Some

important physical properties of fused silica are listed in table 5.1.

Wavelength (nm)

0 500 1500 2500 35000

20

40

60

80

100

Tra

nsm

issio

n(%

)

600

Rayleighscattering

OH absorption

UV absorption

tail

IR absorptionA

tte

nu

atio

n /

(d

B/k

m)

800 1000 1200 1400 1600 18000.1

0.3

1.0

3.0

Wavelength (nm)

Figure 5.1.: (a) Transmission spectrum for a 1 cm thick Suprasil Heraeus sample. (b) Attenu-ation coefficient spectra for fused silica provided from Heraeus.

Figure 5.1 shows the transmission and attenuation spectrum of fused silica. The trans-

mission spectrum is obtained for a 1 cm thick Suprasil sample, which is of high purity

fused silica glass. It is clear from the transmission spectrum that silica has a very low loss

transparency window in the range from 200nm to 2000nm. Transmission in UV range (<

85

5. Material properties

silica

Molecular formula SiO2

Density (g/cm3) 2.20

Expansion coefficient(K−1) 0.54×10−6

n (at 1550nm) 1.44

Tg (°C) 1175

Softening temp.(°C) 1585

Transmittance (µm) 0.18 to 2.2

Drawing temp.(°C) 1900 to 2050

Table 5.1.: Material properties of silica glass.

200nm) decreases due to the presence of absorption bands. These absorption bands arise

due to molecular electronic resonances. The transmission of fused silica also decreases in

the mid-infrared region due to the presence of vibrational resonances.

Apart from these intrinsic absorption processes, the presence of water molecules (i.e.

O-H bonds) and scattering also increases the attenuation coefficient. The sharp dip in

the transmission spectrum at 2700 nm is due to the fundamental absorption band of

O-H (as shown in figure 5.1). The attenuation and transmission spectra also show the

presence of overtone absorption bands of O-H near 1200 and 1400 nm. The lower limit

of the attenuation coefficient for the fused silica is determined by Rayleigh scattering.

If we combine all these attenuation effects, the locally possible minimum attenuation

is obtained at near 1550nm. This is the main reason why most commercial optical

communication fibers operate at a wavelength of 1550nm.

It is important to point out that O-H absorption bands and extrinsic scattering centers

(dust particles) can also be introduced in silica during the stacking and drawing process

of the fiber. The presence of these impurities can be minimized by drawing fiber in a

clean-room environment which will be explained later.

The optical response of a material is determined by its dielectric function. At a specific

wavelength, the real part of the dielectric function (ǫr) is the square of the refractive

index (n) of the material. The dispersion (refractive index variation with respect to

wavelength) of silica can be expressed in form of Sellmeier equation [24]. For practical

applications only the first three terms are important and Sellmeier equation for fused

86

5.1. Dielectric: silica

silica can be written as

ǫr(λ) = n2(λ) = 1 +B1λ

2

λ2 − C1

+B2λ

2

λ2 − C2

+B3λ

2

λ2 − C3

(5.1)

Here, B1,2,3, C1,2,3 are coefficients and λ is the wavelength (in µm). The value of these

coefficients for fused silica glass is given in table 5.21. Using equation 5.3, one can

j 1 2 3

B 4.73115591 · 10−1 6.31038719 · 10−1 9.06404498 · 10−1

C 1.29957170 · 10−1µm 4.12809220 · 10−3µm 9.87685322 · 101µm

Table 5.2.: Sellmeier coefficients of Heraeus Suprasil.

obtain the dispersion curve of the fused silica (figure 5.2 ). The imaginary part of the

400 800 1200 1600

1.44

1.45

1.46

1.47

Re

fra

ctive

in

de

x

Wavelength (nm)

Fused silica

Figure 5.2.: Variation of refractive index with wavelength for fused silica using equation 5.1.

dielectric function of silica, which is related to the absorption losses of silica, is considered

negligible for the wavelength range from 300nm to 1800nm. This is consistent with

the fact that silica is highly transparent in this region. The variation of viscosity as a

function of temperature also plays a very important role in the fiber drawing process of

any material. The viscosity of silica varies smoothly with temperature [57]. Combining

1The value of these coefficients are taken from the data sheet of Heraeus. http : //www.heraeus −quarzglas.de/media/webmedialocal/downloads/broschrenmo/SODatenundEigenschaftenDE.pdf

87


Germanium Silicon

Chemical formula Ge Si

Indirect band gap (eV) 0.66( 291K) 1.12 (300K)

Direct band gap (eV) 0.80 (291K) 4.135 (190K)

Lattice parameter(A0) 5.6579 (298K) 5.4310 (295K)

Linear thermal expansion coeffi.(K−1) 5.90 ×10−6 (300K) 2.92 ×10−6 (293K)

Density g/cm3 5.3234 (298K) 2.329002 (298K)

Melting point Tm(K) 1210.4 1687

dTm/dP (Kbar−1) -1.85 ×10−3 -3.52 ×10−3

Intrinsic conductivity Ω−1cm−1 2.1 ×10−2 (300K) 3.16 ×10−2 (300K)

Intrinsic carrier concentration cm−3 2.33 ×1013 (300K) 2.33 ×1013 (300K)

Table 5.3.: Important physical and electronic properties of Germanium and silicon [44].

all these characteristics of fused silica, it is found to be a excellent material for fiber

drawing.

5.2. Semiconductors

The main objective of this work is to investigate the interaction of light with semicon-

ductor materials in PCFs. Germanium (Ge), silicon (Si) and silicon:germanium (Si:Ge)

alloys were the investigated candidates. In our sample fabrication method, it was difficult

to maintain the composition, phase and homogeneous distribution in Si:Ge alloys. So

we have focused on pure single compound semiconductors like silicon and germanium,

specially germanium.

5.2.1. Germanium (Ge)

Germanium is an indirect semiconductor. It is one of the most widely used semiconductor

for optical devices in the infrared region due to its low loss transparency window from

2µm to 16 µm. Table 5.3 shows some of the important physical and electronic properties

of germanium and silicon.

88

5.2. Semiconductors

The reason for choosing germanium in the present work is because of its lower melting

point in comparison to silicon and silicon (400K below the Tm for silica). Due to this

large difference, PCF structures remain stable during the filling of molten germanium

into the empty holes of PCFs (exact procedure will explained later in section 6.3). In

case of silicon, the melting point is quite close to the softening point of silica. It makes

filling of silicon into the holes of PCFs challenging without changing or even destroying

the PCF structure.

The dielectric function of germanium is calculated from the refractive index (n) and

the attenuation coefficient (k). These parameters can be obtained experimentally by

measuring the optical absorption and reflectivity of the sample. The most common

experimental methods to calculate the dielectric constants of germanium are reflectivity

and ellipsometry measurements. If the reflectivity measurements are carried out over a

sufficiently large range of energies, the Kramers-Kroenig relation can be used to calculate

the real and the imaginary part of the dielectric function of germanium [58] [48]. Using

the ellipsometry measurements, Vina reported the dielectric function of germanium at

different temperatures [59]. In addition to that, interferometric [60] and prism based

methods [61] were also used to determine the dielectric function of germanium. Using

figure 3.13, the optical absorption spectrum of pure germanium is discussed in detail

in section 3.4. For simulation studies in this work, the experimental data of n and k

reported by Philipp and Taft is used to calculate the value of the real (ǫr) and imaginary

(ǫi) part of dielectric function [48]. Using this data the real (ǫr) and imaginary (ǫi) part

of dielectric function can be plotted as a function of wavelength.

400 800 1200 16004.0

4.5

5.0

5.5

6.0

Refr

active index

Wavelength (nm)

Germanium

Wavelength (nm)

400 800 1200 16000

10

20

30

Real part of

dielectric function

Real part

of

?ge

Imaginary part of

dielectric function

0

10

20

30

40

Imagin

ary

part

of

?ge

Figure 5.3.: (a) Dielectric function (real and imaginary) of germanium as a function of wave-length [48]. (b) Refractive index of germanium as a function of wavelength (Dispersion curve ofgermanium) [48].

89


Figure 5.3 shows the dielectric function and refractive index spectra for germanium.

One of most important features of germanium is the temperature dependence of its optical

and electrical properties. Based on the discussions in section 3, we can summarize some

important temperature dependent properties of germanium.

Band gap: Temperature dependence of the indirect energy band gap is shown in figure

3.9 and discussed in detail in section 3.3.2. The indirect energy band gap of Ge decreases

with the increase in temperature, and at room temperature (300K) its value is 0.66eV

[44]. The direct band gap of germanium is 0.80eV at 300K and it also decreases with

increase in temperature [44].

Linear thermal expansion coefficient: The effect of temperature on the lattice con-

stant and thermal expansion coefficient of germanium is shown in figure 3.8 and discussed

in section 3.3.1. The thermal expansion coefficient for germanium is 5.90× 10−6 K−1 at

300K and increases to 7.20 × 10−6 K−1 at 600K. Near the melting point of germanium

the expansion coefficient increase to 8.51×10−6 K−1 at 1000K [44]. Germanium is one of

the few materials which expands on solidification from its liquid phase due to its density

anomaly [45] [46].

Intrinsic carrier concentration: The intrinsic carrier concentration of germanium

increases with temperature (room temperature value is 2.33 ×1013 cm−3 [44]). The

temperature dependence of the intrinsic carrier concentration is shown in figure 3.7 and

discussed in detail in section 3.2.4. The effect of temperature on the carrier concentration

for intrinsic germanium can be approximated by the following empirical relation

n2i = 3.10× 1032exp(−0.785/kT )cm−6 (5.2)

where kT is in eV.

Electrical conductivity: The electrical conductivity of germanium increases with tem-

perature as shown in figure 3.11 and discussed in detail in section 3.3.3. At 300K its

value is 2.1 ×10−2 Ω−1cm−1 [44].

Optical absorption and dielectric function: The effect of temperature on the optical

absorption in germanium is discussed in detail in section 3.4.1.5. A detailed study of

the effect of the temperature on dielectric function of germanium using ellipsometry

measurements was reported by Vina et al. [59]. One of the important conclusions made

by Vina et al. was that both the spectra (for the real and imaginary part of the dielectric

90

5.2. Semiconductors

function) shift towards shorter wavelengths as the temperature is increased. Also the

spectral features (peaks) broaden with increase in temperature.

Refractive index: The temperature dependence of the refractive index for germanium

has been studied by various authors such as Briggs [61], Rank et al. [62], Icenogle et

al. [63] and Lukes [64]. All these authors concluded that for a specific wavelength,

the refractive index of germanium increases with increase in temperature. But there is

disagreement in the value of the temperature coefficient of the refractive index dn/dT

for germanium in these studies. Most of these works were done for wavelengths greater

then 1800nm. Kronfeld experimentally investigated the change in refractive index with

temperature for a pure germanium (prism cut) sample in the spectral range from 1500nm

to 2200nm and temperature range from 80K -460K. Using these measurements the value

of dn/dT can be approximated as 5.7 × 10−4K−1 at 1560nm. For long wavelengths (>

1600nm), this value decrease as the wavelength increases and can be approximated as a

linear constant value beyond 4000nm, as reported by various researchers [64][63] [60]. It

has been reported that the temperature coefficient dn/dT is a function of temperature

and wavelength [64]. Lukes reported that the variation of dn/dT for Ge in temperature

range of 100-500K (for wavelengths > 1970) is nonlinear and not constant as reported by

Icenogle et al. [63]. The spectral dependence of dn/dT is suggested by several researchers

such as Kronfeld. He concluded that for a fixed temperature, dn/dT has a sharp increase

with decreasing wavelength (for wavelength < 1800nm). In addition to these studies for

a crystalline germanium mentioned above, Goldschmidt investigated the dispersion of

amorphous germanium for various temperatures and proposed a new model based on the

dispersion of two oscillators [65].

One empirical way to determine the refractive index of germanium as function of tempe-

rature was proposed by Barnes et al. [66]. He suggested temperature dependent Sellmeier

coefficients which can be used in Sellmeier equation to calculate the dispersion of germa-

nium at different temperatures. It must be noted that these coefficients were calculated

under the approximation that dn/dT is a linear constant in this range. If we write the

sellmeier equation for germanium as

n2 = A+Bλ2

λ2 − C+

Dλ2

λ2 − E(5.3)

91


Then the coefficients A,B,C,D,E are function of temperature

A = −6.040× 10−3T + 11.05128 (5.4a)

B = 9.295× 10−3T + 4.00536 (5.4b)

C = −5.392× 10−4T + 0.599034 (5.4c)

D = 4.151× 10−4T + 0.09145 (5.4d)

E = 1.51408× T + 3426.5 (5.4e)

here, T is in Kelvin. The value obtained using this method shows a mean square deviation

of 0.00142 in comparison to the reported experimental value of the refractive index at

297K.

Pressure also affects the optical properties of germanium. The effect of pressure on the

direct and indirect energy band gap of germanium was studied by various researchers such

as Aadler and Erlbach [67], Cardona et al. [68] [60], Paul and Warschauer [69], Slykhouse

and Drickamer [50]. The dual behavior of the pressure coefficient of the energy band gap

and its effect on the optical absorption has already been discussed in section 3.4.1.6. With

increase in pressure first the energy band gap increases and has a pressure coefficient of

≈ 8× 10−6 eV cm2/kg. This positive pressure coefficient changes sign at around 50,000

atm. Further increase of pressure (above 50,500 bar) decreases the energy band gap with

a pressure coefficient of ≈ 5× 10−6 eV cm2/kg.

5.3. Nobel metals: gold

Gold is a very important and widely used material to investigate the light matter inter-

actions specially in plasmonics. It shows great resistence to the corrosion and oxidation

in air. Some important physical properties of gold are listed in table 5.4. The reason

of choosing gold in this work is that it has a relatively low imaginary part of dielectric

function (which is related to optical absorption of Au) in comparison to other metals.

Although, copper and silver has relatively low imaginary part of dielectric function but

they have higher diffusion rate in silica specially at higher temperatures.2 The dielectric

function of the metal can be described by Drude model (for metals without sharp intra-

band transitions). Gold shows intraband transition at wavelengths <600nm, a modified

2This property is very useful in context of this work.

92

5.3. Nobel metals: gold

Germanium

Chemical formula Au

Linear thermal expansion coeffi.(K−1) 14.0 ×10−6

Density g/cm3 5.3234 (298K)

Melting point T(°C) 1063

Table 5.4.: Important properties of gold [70]

model including two additional resonances was used to calculate the dielectric function

(ǫ) [71].

Imagin

ary

part

ε Au

Real part

of

ε Au

Wavelength (nm)

Re(εAu)Imag(εAu)

400 800 1200 1600-120

-100

-80

-60

-40

-20

0

0

4

8

12

Figure 5.4.: Dielectric function (real and imaginary) of gold(Au) as a function of wavelength[71].

93

Chapter 6Fabrication, characterization, measurement

and simulation techniques

6.1. Fiber fabrication: stack and draw method

The fabrication of high optical quality fibers is an integral part of this work. It is of

utmost importance that these fibers are designed and fabricated in such a way that

not only their transmission characteristics cover the desired wavelength range (400nm-

1600nm) with low loss, but they also provide an efficient way of allowing the interaction

of material (metal or semiconductor) with light. Photonic crystal fibers (PCFs) have

proven to be an ideal candidate to study these interactions. The most common way of

fabricating these fibers is the stack and draw method.

6.1.1. Photonic crystal fiber

In this work, photonic crystal fibers (PCFs) are made of fused silica. This section briefly

describes the basic process to fabricate a standard PCF. The stack and draw method to

fabricate PCFs involves two stages,

• the stacking process (in which the preform is made

• the fiber drawing process (in which the preform is drawn into actual fiber).

Stacking process:

95

6. Fabrication, characterization, measurement and simulation techniques

Figure 6.1 shows various steps of the stacking process. The first step is to draw capillaries

of required dimensions from a commercially available fused silica tube or rod(Heraeus).

The ratio of their outer diameter (OD) to the inner diameter (ID), i.e OD/ID is very

important. In our case, the typical length of the rod is ≈ 1.5m and OD ≈ 20mm. This

tube is then drawn down to capillaries of OD ≈ 2mm, using a high temperature resistance

furnace.

Step 1: Drawing capillaries using cane tower.

The furnace contains a graphite element of cylindrical shape, which can create a uniform

heat zone of up to 2100°C inside the furnace. The furnace is continuously purged with

Argon(Ar) gas. This purging also helps to prevent impurities (O-H groups, dust particles)

from moving into the molten glass during the drawing process. The furnace surroundings

are water cooled, allowing a fast and precise control over the operating temperature.

Figure 6.2 shows of the cane tower. The fused silica tube (OD = 20mm) is first fitted into

the preform chuck holder at top of the tower. The furnace is heated up to a temperature of

≈ 2050°C and then the silica tube is moved into the furnace. The silica tube melts inside

the furnace and slowly forms a ”drop”-like structure, which comes out from the lower

opening of the furnace. This process is called drop-off. As the drop-off moves downward

due to gravity, a slow feeding (rate of movement of silica tube into the furnace) of silica

tube is started.

Then, the temperature of the furnace is decreased in a very controlled manner (a rate

of 5K per step) down to 2000°C. The drop-off part is then cut off and the drawn tube

is clamped in between the conveyor belts of tractor wheels which are located below the

furnace (as shown in figure 6.2). The tractor wheels pull the tube at a specific rate which

is referred to as drawing rate in further discussion. This drawn tube is referred to as

’capillary’. Next, the drawing parameters, i.e. the feed rate of silica tube, the drawing

rate of capillary and the temperature are varied to achieve the required OD/ID ratio of

the capillary. The diameter of the tube is monitored using a laser based device called

the ’laser mike’. The value of the feed rate of silica tube and tractor pulling speed of

capillary is related to the volume of glass being fed in and pulled out of the furnace. At

any given time, the in and out mass of the glass must be conserved. Hence, the feed

rate vin, OD of the silica rod din, pulling speed of the tractor vout and OD of the drawn

capillary dout are related by the conservation of total volume,

vind2in = voutd

2out (6.1)

96


Step 1: Drawing capillariesof required diameter and length.

Step 2: Stacking capillaries.

Step 3: Putting stacked capillariesin silica jacket tube.

Silica tube (from glass manufacturer)

~20mm

~2 mm

Figure 6.1.: Schematic diagram showing various steps for stacking procedure in PCFfabrication.

97


Preform chuck

Stackedpreform

Furnace

Tractorwheels

Diametermonitor

cane

Cane Tower

Canepreform

Spoolingunit

Tractorwheel

UV lamp

Coatingunit

Preform chuck

Furnace

Diametermonitor

Coated fiber

Fiber Tower

Cuttingpoint

(a) (b)

Figure 6.2.: (a)Cane drawing tower. (b) Fiber drawing tower.

98


Based on this equation and value of the diameters involved, the feed rate is calculated

for a given tractor speed or vice versa. Once the process is stabilized for a given output

diameter, the temperature is varied to obtain a circular and stable structure of the

capillary. The change in temperature also changes the OD and ID of the capillary, which

necessitates some fine adjustments in the drawing speed to compensate for the change.

Additional pressure can also be applied inside the tube to attain a specific OD/ID ratio.

These capillaries of length ≈ 1.20 meters are then collected by cutting them below the

tractor. Step 2: Stacking of capillaries.

After obtaining the required number of capillaries for a specific fiber structure, these

capillaries are assembled together in a specific geometry on a custom made rig. This

process is called the stacking of capillaries. The geometry of the stacked capillaries can

be considered as an upscaled structure of the final fiber design. The central region of the

stack can have solid silica rods (solid core PCF design), capillaries of different OD/ID

ratio (polarization maintaining (PM) PCF) or hollow region (hollow core design). After

finishing this stage of stacking, these capillaries are tied together with copper wires.

Step 3: Fitting the stack in jacket tube.

The stack is then inserted into a silica jacket tube and copper wires are removed. The

ID of the jacket tube is chosen in such a way that the stacked capillaries fit just inside

the tube, while the OD is chosen with consideration of the scaling factor in the drawing

process. In some cases, additional silica rods are also inserted to fill to avoid instabilities

during the drawing. The final stacked structure is called ’preform’. The jacket tube

contains a small slot, which is 15 cm below from one of the end face. This end face of

the stack is then fused completely and then vacuum connector can fixed over this slot.

Fiber drawing process:

The second stage is the drawing process. The number of steps involved in drawing

depends on the final scaling ratio from preform to fiber and also on the stability of the

structure. For PCF, stable drawing can be achieved with a two step drawing process.

The most challenging aspect of PCF drawing is to maintain the structure of the preform

in such a way that the final fiber has the desired dimensions, i.e. correct diameter and

pitch ratio (d/Λ). Figure 6.3 shows different steps for drawing a fiber from the stacked

preform in PCF fabrication process.

Step 1: Drawing of canes.

99


Stacked preform for canes

Step 1: Drawing of canes.

Step 2: Putting cane in asuitable silica jacket.

Step 3: Finally drawing fiberof suitable parameters.

Photonic Crystal Fiber

~20cm

~1.7cm

~10mm

Figure 6.3.: Schematic diagram showing various steps for fiber drawing procedures.

100


In the first step of the drawing process, a miniature version of the preform called ’cane’

is drawn from the preform using the cane tower. The drawing of canes from the preform

is similar to drawing capillaries from silica tube.

Once this process is stabilized the temperature is varied to achieve a stable structure

inside the cane. It is important to mention that decrease in temperature also increases the

OD of the cane. To compensate this, the drawing temperature, feed rate and the drawing

speed are carefully adjusted to achieve a stable cane structure. Vacuum is applied to the

preform to overcome the gaps among the capillaries and the gaps between the capillaries

and jacket tube. The cane structure is regularly checked using an optical microscope

during drawing, and suitable adjustments are made in the drawing parameters. Once

the desired structure is achieved, several canes of length ≈ 1.2m can be collected from

the drawing of a single preform.

Step 2: Drawing of fiber from cane.

In the second step of this process, the actual fiber is drawn from these canes. A cane is

fitted inside a suitable silica jacket tube, with a maximum air gap of ≈ 100µm. Figure

6.2 shows the schematic design of the fiber drawing tower with some of its essential

components. Components like furnace, diameter control unit and preform chuck are the

same as in cane tower. The fiber tower has a coating unit and a UV lamp below the

diameter control. The coating unit is used to coat the fiber with a polymer layer, followed

by the UV lamp which is used to polymerize the polymer layer on the fiber surface. This

polymer layer provides mechanical stability to the fiber and also protects the fiber from

any chemical contamination and water moisture. Below the UV lamp there is a set of

wheels, which controls the drawing speed of the fiber. They also provide the required

’tension’ for stable drawing. For a stable drawing process the value of the tension must

remain in between 100 - 400g. In the end, a foam spool is used to collect the fiber .

The typical OD of the fiber is ≈ 200µm. The temperature, feed rate and drawing speed

are continuously monitored and the fiber structure is checked in an optical microscope.

Pressure and vacuum are also applied to keep the holes open and to close the interstitial

gaps inside the cane. It should be noted that a cane of OD ≈ 2mm can produce ≈1km length of fiber with OD ≈ 200µm, which means that a single cane can produce

several set of fibers, which have different physical parameters but similar geometry. The

characteristics of these fibers are investigated using scanning electron microscope (SEM)

and transmission measurements. Figure 6.4 shows the SEM images for some of the most

common PCF structures.

101


10µm 15µm

10µm5µm

(b)(a)

(d)c( )

Figure 6.4.: (a) Solid core endlessly single mode PCF. (b) Hollow core PCF. (c) Polarizationmaintaining (big hole) PCF. (d) Polarization maintaining (small hole) PCF.

6.1.2. Modified step index fiber

Besides solid core PCF, another fiber geometry is also investigated in the present work.

This geometry is a modified version of the conventional step index fiber, in which an

empty hole of a specific dimension is introduced adjacent to the GeO2-doped silica core

of a conventional step index fiber. This fiber is addressed as modified step index fiber

(MSIF) in this work. The fabrication process of the MSIF is similar to solid core PCF.

The MSIF fiber can be fabricated using two different approaches. In the first approach,

one GeO2-doped silica rod, five silica rods and a capillary are stacked inside a silica

jacket. All of these capillaries and rods are of same OD ≈ 2mm.1 They are stacked in a

geometry where the GeO2-doped silica rod is kept in the middle, and the five rods and

one capillary form a ring around the GeO2-doped rod. From this preform, the MSIF cane

and eventually the MSIF fiber can be drawn using the fiber drawing procedure discussed

in section 6.1.1. Due to the asymmetry of the structure (position of hole), maintaining

a circular hole and circular shape of the core is a significant challenge in this design and

1The GeO2-doped silica rod of OD 2mm is drawn from a 16 mol.% GeO2-doped silica rod of OD 25mm.The 20mm GeO2-doped silica rod is supplied by commercial manufacturer Hereaues.

102


drawing process. Therefore, another approach based on drawing a template structure is

used to fabricate the MSIF. Figure 6.5 shows various steps of the fabricating MSIF using

a template approach. These steps are discussed as follows:

Step 1: In the first step, seven capillaries of an OD ≈ 2.5mm are stacked together inside

a 10mm OD jacket tube, the geometry of which is shown in figure 6.5. This preform is

used to draw a template cane. The template cane has seven holes, six in a ring and one

in the center.

Step 2: The template preform is fixed in the preform chuck of the cane tower. Template

canes are drawn from this preform using the cane drawing procedure discussed in section

6.1.1.

Step 3: In this step, one of the template cane is used to made a preform from which the

MSIF canes are drawn. A GeO2-doped silica rod is inserted into the central hole of the

template cane. The GeO2-doped silica rod of desired OD (depending on the ID of the

hole in the template cane) is drawn from a commercially available 16 mol.% GeO2-doped

silica rod (OD = 25mm) on the cane tower. The other five holes in the template cane

are filled with silica rods. This filled template cane is then inserted into a silica jacket

tube of the desired OD, which forms the final preform.

Step 4: The preform (made in step 3) is then fixed on to the preform chuck holder of

the cane tower and MSIF canes of suitable OD are drawn from this preform. Depending

on the required diameter of the empty hole of MSIF cane, pressure can also be applied

to keep the hole open (or to change its diameter) in the drawing process.

Step 5: The MSIF cane is fitted inside a suitable jacket tube.

step 6: In the final step, this jacketed cane is fitted into the preform chuck holder of the

fiber tower. Using the fiber drawing process discussed in section 6.1.1, MSIF of desired

core size and OD can be drawn from this cane. MSIFs of same core size and OD with

different hole sizes can be drawn from one cane. In this structure, the diameter of the

GeO2-doped silica core determines the single mode transmission range of the fiber. A

core of diameter ≈ 1.10µm provides a single mode MSIF from 400nm to 900nm.

This simple structure is not only very flexible in design but also works as an excellent

host to investigate the metal and semiconductor interaction with light when this empty

hole is filled. Due to the presence of only one single hole in the structure, the adjustment

of drawing parameters are relatively easy.

103


Seven capillaries stacked as preform for the template.

Step 1: Drawing of the template cane fromthe stacked preform.

Step 2: Five holes of template cane are filledwith silica rods, the central hole with a GeO

doped silica rod. The template caneis fitted in jacket tube.

2

Step 3: Drawing of the MSIF cane.

Step 4: Putting the cane into a suitablesilica jacket tube.

Step 5: Drawing of MSIF from thejacketed cane.

Modified step index fiber (MSIF)

~ 10mm

~ 1.7mm

~ 8mm

~ 1.7mm

~ 8mm

Figure 6.5.: Schematic diagram showing various steps of drawing a modified step index fiber(MSIF).

104

6.2. Post-processing techniques


Post-processing of the PCF/MSIF is widely used tool to locally alter the geometry of the

structure. This process helps to fabricate several in-fiber devices. Fiber characteristics

(e.g. dispersion, air filling fraction, mode area, birefringence) can be modified by post-

processing of PCF/MSIF. Post- processing involves locally inducing heat (along the short

length of fiber) up to the softening point of silica, and then modifying the structure by

pulling, pressurizing or ’cooking’ the holes. There are several methods of providing heat

to the fiber in a very controlled manner, such as by using the butane flame, CO2 laser or an

electrical filament (using the filament of conventional fiber splicing machine). Depending

on the type and precision of the final structure, one of the these techniques of providing

heat can be used to perform the post-processing.

In this section three basic mechanism of post-processing are explained. It is easy to

understand these mechanisms for a single capillary instead of a PCF. The basic process

of post-processing of a single capillary and a PCF is equal, only some fine adjustment

of parameters are required for post-processing a PCF. These three basic mechanisms of

post-processing (for a single hole capillary) are shown in figure 6.6.

Tapering

Hole-inflation

d <d3

L >L3

L L L2≈

d =02Hole-

collapsed

L L1≈

d >d1

Figure 6.6.: Schematic diagram showing three basic mechanisms of fiber post processing.

The first process is the hole inflation. In this process, the diameter of the air-hole can be

105


increased along the axis of the hole using heat and pressurizing the hole. It is important

to have smooth transition between the inflated and un-inflated region of the fiber. In

the second mechanism, the heat is used to completely collapse the hole of the capillary.

This mechanism is called the hole-collapsing. The most important and widely used post-

processing mechanism is tapering. In this process, the diameter of the hole is tapered

down (decreased) in a very controlled manner along the axis of the hole. Apart from

these three basic steps of post processing, periodic structures e.g fiber gratings, rocking

filters can also be created in the fiber structure.

6.2.1. Flame based tapering rig

One of the most common and conventional methods of post-processing is to heat the

fiber by using a flame of butane and oxygen. Figure 6.7 shows a schematic diagram of

Argon pressure(Optional)

Fiber

Butane + Oxygenflame

xyz-translationalstage

Figure 6.7.: Schematic diagram of fiber tapering rig.

the tapering rig. First, the polymer coating of the fiber is stripped-off from its outer

surface because any small section of coating on the surface of the fiber would be burned

during the post-processing and would contaminate the surface. The fiber is then cleaned

using acetone (CH3COCH3) and isopropanol (C3H8O). Cleaning not only removes the

residual coating, but also removes any major dust particles on the surface. Using fiber

holders, the cleaned fiber is then clamped onto two 3-dimensional translational stages.

These stages can be moved along the fiber axis. A commercial software (LabView) is

used to precisely control the movement of these stages on the rails. The flame is also

fixed on a movable stage. The flame of butane and oxygen is adjusted just below the fiber

to create a uniform heat zone across the fiber (at the position where flame is touching

the fiber). By sweeping the flame at a specific rate, the heat zone is moved along the

106


axis of fiber. The ratio of butane and oxygen (1:2) is kept in such a way that a uniform

heat zone is maintained. The approximate size of the flame is 2-5mm.

To have adiabatic transition between the tapered and un-tapered region, the sweep rate

is fast (20mm/sec) and the flame is moved back and forth several times across the specific

length of the fiber [72]. The number of turns and the sweep rate are determined by the

final waist diameter. The tapering is a two step process; in the first step, the flame sweeps

back and forth at a constant distance called the initial heating distance. Simultaneously,

the two stages, on to which the fiber is clamped, is pulled outwards at a constant speed.

This phase forms the conical transition between the un-tapered and tapered part of the

sample. The length of the initial heating zone defines the slope of the transition. The

ratio of the initial heated distance and the final tapered waist length determines the

diameter at which the second phase starts. If the two distances are similar, almost the

entire diameter reduction happens during the first phase, resulting in a long transition.

The second phase is responsible for the waist section. By slightly increasing the flame

sweep distance with each turn, reversal point of the flame is kept up with the fiber

pulling speed. It allows to maintain a homogeneous diameter along the whole section of

the waist.

For the hole-inflation process, a low sweep rate is used so that there is enough time for

the hole to inflate. The hole inflation is achieved by applying a specific pressure to the

hole while sweeping the fiber with the flame. The value of the pressure required to inflate

the hole depends on the hole diameter and the OD of the fiber. Both the tapering and

hole inflation of the PCFs were studied in detail by Wadsworth et al. [72]. The value of

critical pressure which is required to open a hole in a silica fiber has been approximated

as

Pst(bar) = 6/d(µm) (6.2)

here, d is the diameter of the hole in µm. This equation suggests that at softening

temperature of the silica, pressure of 6bar is necessary to keep a 1µm hole open. The

value of actual pressure also depends on the actual structure of fiber. The equation 6.2

also indicates that the critical pressure increases as the hole diameter decreases.

6.2.2. Splicer

Other than the flame based tapering rig, another way of efficient post-processing of

PCF/MSIF is to use a conventional splicing machine. In a splicing machine, the efficient

107


heat zone across the fiber is created by electrical filament. A Vytron splicer is used to

perform the post-processing of PCF/MSIF/capillary. The basic method for tapering,

hole collapsing and hole inflation remains the same as discussed in the previous section.

If the required length of the post-processed region is very small(1-2mm), then splicer

based post-processing has advantage over the flame based post-processing because of

small local heating region of filament. The precise control over the temperature of the

heat zone, fast processing and simultaneous assessment of the processed region by using

the splicer camera, are very useful to make several samples in a short amount of time. A

schematic diagram of the Vytron splicer is shown in the figure.

Tungsten

Two fibers tobe spliced

Stages for fiberalignment

Figure 6.8.: Schematic diagram of a filament based splicer.

6.2.3. Selective hole treatment

One of the most important features of PCF post-processing techniques is modifying the

diameter of specific holes in the PCF cladding. One of the examples of selective hole

treatment is to open one hole adjacent to the core of the fiber and collapse the rest of

the holes. It is achieved by using a combination of hole-inflation and hole-collapsing

processes as discussed in sections 6.2 and section 6.2.1. Figure 6.9 schematically shows

the steps used to selectively open a single hole (adjacent to the core) and collapse the

rest of the holes of the cladding. First fiber is stripped-off from the polymer coating and

cleaned properly. The different steps shown in figure 6.9 are as follows:

Step 1: First, a drop of polymer glue is dropped on one of the cleaved end faces of the

108


Layer of polymer glue on one end face of fiber andfocus laser to polymerize glue at a specific hole.

Fill polymer glue in rest of the holes using vacuumat other end and harden the glue using UV light.

Cleaving the end face to re-open the blocked hole

Collapsing all the holes at the second end face using heat

Hole inflation of selected hole using Argon pressure and heat.

Processed fiber with single hole opened at one ofthe end face.

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6

Laser beam

:

Argon pressure

.

.

Figure 6.9.: Schematic diagram showing various steps involving selective hole openingprocedure.

109


PCF. The glue consist of a hybrid polymer (Okmcore b59)2 mixed in a solvent (Ormothin)

in a ratio of 1:3. The viscosity of the final combination is related to hole size, for large

holes (10 µ the ratio of 1:2 is used) . A laser is then focused very carefully on the selected

hole (which is selected to remain open in the end). Due to two photon polymerization,

the glue at the focused spot of the laser is polymerizes and becomes hard. The two-

photon polymerization requires a threshold energy, which is used to achieve a small area

of the hard glue.

Step 2: In the second step, the glue is pumped into the rest of the holes of PCF excluding

the selectively closed one. This is achieved by dipping the end face (which has a layer

of glue) of the fiber into a small container filled with glue, and then applying vacuum to

the holes of PCF from the opposite end face. The length of the filled holes is typically

8-10 cm. The hole which is required to remain open has a layer of hard glue on its end

face, while no polymer glue enters into this hole. Once the other holes are filled with the

glue, UV light is illuminated over the entire filled section of the fiber in order to harden

the glue.

Step 3: A very small length of the glue filled section is cleaved from the end face where

all the holes are filled with hardened polymer glue. It results into a PCF end face in

which one hole is completely open and the rest has hard glue inside. Now this fiber is

now ready to be post-processed using either a flame based tapering rig or splicer.

Step 4: In the fourth step, all the holes of the opposite end face of the fiber are collapsed

by using the heat from a butane and oxygen flame of tapering rig or by the filament of

the splicer.

Step 5: Using argon gas, sufficient pressure is applied (depending on the diameter of

the opened hole) from the glue-filled side of the fiber. Somewhere between the collapsed

section and the glue-filled section, the hole inflation process (as discussed in detail in

section 6.2 and 6.2.1) is performed on the fiber . As all holes except the selected one are

blocked by the long channels of glue, these blocked holes do not allow the pressure to

arrive at the section of the fiber where the hole-inflation is being performed. The selected

hole (which does not have glue) becomes pressurized and remains open during the hole-

inflation process. It is important to choose the value of the pressure very carefully; the

pressure must be enough to keep the hole open or at least slightly inflated to ensure a

smooth transition between the processed and unprocessed section.

2From micro resonant technology.

110


Step 6: The inflated (processed) region is checked under the microscope, and the position

of inflated section is marked carefully. In the final step, the fiber is cleaved exactly at

the position where the hole-inflation has been performed. This results in a fiber end face,

where all the holes are collapse except one. The glue-filled part can now be removed

(cleaved-off) to use the sample for further investigation.

10µm

10µm 10µm

10µm

(a) (b)

(d)c( )

Figure 6.10.: Optical microscope/SEM pictures of end face of ESM PCF at various stages ofselective hole inflation. (a) Original ESM PCF.(b) With a layer of glue (step 1).(c) After focusinglaser light on the a selected hole (step 2) (b) After post-processing the end face with one openhole (step 6).

Figure 6.10 shows the end face of an ESM fiber at various stages of selective hole inflation

process. These type of samples are very useful in the context of this work.

111


6.3. Filling techniques in PCFs

This section deals with the fabrication techniques of structures in which light can interact

with semiconductors or metals. The most obvious way to investigate this interaction is

to fill the hollow air channels in the PCF/MSIF with semiconductors and metals. The

possible candidates of filling material are Si, Ge and Si:Ge as semiconductors, and gold

as metal (section 5). In this section, two different filling techniques are discussed (i)

direct drawing and, (ii) post fiber drawing filling techniques. In direct drawing the

metals/semiconductors filled structures are fabricated by filling these materials inside

the holes of a cane, and then draw the cane into filled PCF/MSIF structures. In post

fiber drawing filling technique, first fiber is drawn and then the holes of PCF/MSIF are

filled with metals/semiconductors.

6.3.1. High temperature vacuum or pressure cell

Schmidt et al. has shown a very unique and efficient method to fill the holes of PCFs with

metal by pressurizing the molten metal into the hollow channels of the PCFs [7] [73]. As

discussed in section 5.2, silicon and germanium are two possible candidates for filling the

holes of PCF/MSIF. The melting point of silicon is close to the softening point of silica

(1600°C), which indicates that the filling of silicon is extremely challenging when using

the approach of high temperature pressure. Therefore, the best candidate is germanium,

with its melting point at 940°C.

pure germanium is used as a filling material in this work. (99.9999%). The most im-

portant challenge when filling germanium is that the filling process must be done in a

vacuum or in argon atmosphere since molten germanium easily oxidizes in the presence

of oxygen. A modified version of the pressure cell is designed to fill the pure germanium

into PCF.

Figure 6.11 shows the schematic diagram of the pressure cell used to fill germanium in

the empty holes of PCFs. The pressure cell consists of a pure fused silica tube of OD

≈ 15mm and ID ≈ 8mm. The length of the tube is typically 70 cm. The silica tube

is chosen for its mechanical stability, transparency (it helps to view the filling process

with eye) and property of not been highly reactive to germanium. It is important to

mention that these tubes are not designed for a combination of high pressure and high

temperature applications.

112


Oxygen + Butaneburner

Guidingceramic tube

Fiber

Viciconnector

Molten germanium

Inletvalve

Exhaustvalve

Argon Argon

Silicatube

Glue

Heat zonedue to burner

Figure 6.11.: Schematic diagram of Ar purged high temperature pressure cell used to fill ger-manium in hollow channels of PCF/MSIF.

This fused silica tube is cleaned by using acetone and isopropanol to remove any dust

impurity at the inner surface of the silica tube. Both end faces of the silica tube are

attached to pressure connectors, using a two component epoxy glue3. This epoxy glue

must have the capability to sustain high pressures ≈ 70bar and temperature ≈ 100°C

after hardening. The pressure connector on one of the end faces has two separate channels

(openings). One channel (which is aligned with the axis of tube) is used to insert the

PCF/MSIF. The other channel is used to purge or pressurize inside the silica tube, using

Ar gas. The pressure connector attached with the opposite end face is single channel

and used to flow out the argon gas. Pressure connectors at both ends of the tube have

valves, which can be opened or closed to control the flow of Ar gas. The inlet channel of

the pressure-connector can be further attached to an Ar gas cylinder or vacuum pump.

The Ar gas cylinder can supply a maximum pressure of up to 200bar. A small piece of

pure germanium is inserted into the silica tube. Few sequential cycles of purging Ar and

creating vacuum are performed to remove any content of oxygen and other impurities.

The pressure cell is continuously purged by a constant flow of Ar gas. The entire system

is in the horizontal orientation.

The PCF is stripped from its polymer coating and cleaned by using isopropanol and

acetone. The length of the PCF is ≈ 70cm. The PCF is then inserted into the pressure

cell through the opening of the pressure connector with the help of guiding ceramic tube.

3Pattex Stabilit.

113


It is pushed inside the pressure cell very carefully, so that its end face does not get

scratched by the wall of the pressure connector or silica tube. Peek connectors from the

company Vici are used to fix the fiber into the pressure connector. The Vici connector

must be able to hold pressure up to ≈ 60bar. The fiber is fixed at a length so that

it just touches the piece of germanium. A flame burner is then used to create a heat

zone around the germanium piece. The burner uses a combination of butane and oxygen

to produce a flame which can create a heat zone (temperature ≈ 1000°C) which is hot

enough to melt the germanium. Ar gas is still continuously purged during this process

to avoid any oxidation of germanium. When germanium reaches to its molten state, the

fiber is slightly moved to push the end face of germanium inside the molten germanium

drop. The Vici connector is then tightly screwed and the exhaust valve is also closed.

A required pressure depending on the hole diameter is then applied inside the pressure

cell (section 6.3.3). This high pressure pushes the molten germanium inside the holes

of the PCF/MSIF. The molten germanium inside the PCF/MSIF solidifies at a certain

distance from the end face where the temperature is low. The pressure is applied for 1-5

minutes depending on the hole diameter and filling material (section 6.3.3). In this case,

the pressure is applied for 2 minutes. After that, pressure is released by opening the

exhaust valve but the constant purging of Ar gas is still maintained. The Vici connector

is unscrewed and fiber is then moved out of the pressure cell very carefully. The flame is

switched off and germanium is allowed to cool down. It should be noted that the filled

section is around 1-2cm long, depending on the length of the heat zone across the fiber.

The value of the pressure used to push the germanium in holes of PCF depends on the

diameter of the holes and the viscosity of the molten germanium. A argon gas pressure

of nearly 45bar is used to fill a hole of 1.6µm (section 6.3.3).

Further characterization of these germanium-filled structures are performed under the

optical microscope and scanning electron microscope (SEM). Figure 6.12 shows the SEM

images of Ge-filled ESM PCF structures, which were fabricated using the pressure cell

filling technique.

6.3.2. Splicing technique

Another technique of filling is based on the use of a conventional fiber splicer.(section

6.2.2) This technique is very flexible and can be used to fill various materials inside the

holes of PCF/MSIF. Figure 6.13 shows various steps of the splice filling technique to fill

114


10 mμ

2 mμ

10µm

(a) (b)

Figure 6.12.: SEM image of some germanium-filled ESM structures using pressure cell tech-nique. (a) completely filled (first ring). (b) Single hole filled ESM PCF.

a MSIF with germanium. This procedure requires a capillary with same OD as MSIF,

referred as ’filling capillary’. The lower limit for the ID of this capillary depends on the

minimum size of the material piece which can be easily pushed inside the capillary. The

upper limit of the ID is determined by the wall thickness of capillary, which can sustain

high pressures due to lower force on the wall (up to ≈ 400 bars). A capillary of OD ≈200µm and ID ≈ 150µm is used in this work. For germanium it is difficult to acquire

wires of diameters below ≈ 150µm. Therefore, Ge-filled silica capillaries of OD ≈ 120µm

and ID ≈ 80µm were fabricated for this process. These capillaries are fabricated using

the filling technique discussed in 6.3.1. As the ID of capillary is quite large, vacuum is

used at the opposite end of the capillary.

Various steps of the filling procedure are as follows,

Step 1: In the first step the filling capillary (OD = 200µm, ID = 150µm) and MSIF (d

=1µm) are stripped-off and cleaned. Cleaning of the fiber and capillary is very essential

in this technique, as dust particle or coating can damage the spliced region.

Step 2: The end faces of both capillary and MSIF cleaned fiber are cleaved in such a

way that they have a flat and straight end face.

Step 3: The cleaved fiber and capillary are then fixed in the fiber holder of the splicer. As

the purpose of the splice is to join MSIF and capillary mechanically, manual alignment is

sometimes preferred over the automatic alignment of splicer. Splicing is then performed

to connect two end faces together. ( splice parameters in appendix A)

Step 4: The filling capillary is cleaved, so that only a length of 6cm remained attach

with MSIF. A small piece of Ge-filled capillary (OD = 120µm, ID = 80µm, length = 5-

115


Pressure

Filling capillary

Fiber polymer coting

Straight and flat cleave

Splicing-1

Ge filled capillary

Splicing-2

Furnace

Step 1

Step 3

Step 2

Step 5

Step 4

Step 6

Modified step index fibercore

hole

Figure 6.13.: Schematic diagram showing various step for filling the hole of MSIF withgermanium.

116


10mm) is then inserted inside the filling capillary from its un-spliced end. This capillary

is carefully pushed towards the spliced end and placed barely touching the end face of

MSIF. This Ge-filled capillary serves as the source of material which is to be filled inside

the MSIF.4

Step 5: A second splice is performed to attach another piece of filling capillary ≈ 200µm

and ID ≈ 150µm with the un-spliced end of the first filling capillary. A second capillary

is attached, only to produce a long section of filling capillary behind the spliced region

(between filling capillary and MSIF).

Step 6: In the final step, this entire sample is moved inside a vertical/horizontal furnace

and kept inside the heat zone in such a way that the spliced region between MSIF and

filling capillary lies in the middle of the heat zone. The other end of the capillary is

attached to the pressure connector using a Vici connector. The pressure connector can

be connected to vacuum pump or pressure supply.

First, a vacuum is applied into the ’filling capillary’ for 3 minutes followed by the purging

of Ar gas in the capillary for 30 seconds. This process is repeated four times to remove

any content of oxygen. After the purging cycles are over, Ar is kept inside the filling

capillary (ensuring a very low pressure ≈ 0.1bar). This is done to prevent the presence

of oxygen in capillary. The furnace temperature is then ramped up to the 1000°C (above

the melting point of germanium). Once the melting temperature of Ge is reached, the

vacuum is stopped and the pressure is applied inside the filling capillary using Ar gas.

The pressure pushes the molten germanium inside the MSIF and is applied for 10-15

minutes. After that, the pressure is released and the furnace temperature is decreased

to room temperature. The sample is then removed from the furnace and filling capillary

is cleaved off from the Ge-filled section .

The splice filling method is relatively safe as the volume of the total gas inside the fiber is

very small. Figure 6.14 shows the SEM image of germanium-filled PM PCF and MSIF,

which were fabricated using splice filling technique. For PM PCF only two small holes

were filled.

4For gold filling, commercially available gold wires are inserted in filling capillary instead of the Ge-filledcapillary.

117


5µm 2 mμ

Ge-wire

core(a) (b)

Figure 6.14.: SEM image of Ge-filled fibers fabricated using splice filling technique. (a) PMPCF (b) MSIF.

6.3.3. Direct drawing method

The filling techniques discussed so far ( in section 6.3.2 and 6.3.1) are used to fill an al-

ready drawn PCF. Another method of filling requires direct drawing a metal/semiconductor

filled fiber structures. This process is very suitable for materials which have a very low

diffusion rate into the silica at the drawing temperature of PCF (≈ 2000°C). Figure 6.15

shows various steps to draw an Au-filled MSIF. Different steps are discussed below:

Step 1: First, a MSIF cane of desired parameters is drawn using the cane drawing

process explained in section 6.1.2. The empty hole of this cane is then filled with gold

with the help of vacuum.

Figure 6.16 shows the schematic diagram for filling the cane of MSIF with gold. Let us

consider that the cane has an OD ≈ 1.7mm, hole diameter ≈ 90µm and length ≈ 1.2m.

First, a silica jacket tube (OD ≈ 6mm, ID ≈ 2mm and length ≈ 1m) is cleaned with

acetone and isopropanol. A commercially available gold wire5 of diameter 1.7mm and

length ≈ 3cm is inserted into the jacket tube. The wire is placed at a distance of ≈ 5cm

from one of the end faces of the jacket tube. The outer surface of this cane is then cleaned

with acetone and and isopropanol. This cane is then inserted into the jacket tube from

the opposite end face. The cane is placed inside the jacket tube in such a way that it just

touches the gold wire. The opposite end face of the cane coming out of the jacket tube

(due to its longer length than jacket tube) is connected with the vacuum pump using a

pressure connector. This pressure connector is attached with the cane, either by using

a heat shrink tube or two component epoxy glue. The whole structure is then placed

5Goodfellow, www.goodfellow.com

118


Step 1: Filling gold in the single emptyhole of the MSIF cane.

Step 3: Drawing of gold filled MSIFusing suitable drawing parameters.

Modified step index fiberfilled with gold

MSIF cane (OD 1.7mm, hd 90 m)≈ µ≈

Step : Putting gold filled cane in toa suitable jacket tube.

Figure 6.15.: Schematic diagram showing various steps of direct drawing based filling method.The diagram shows the steps for the fabrication of Au- filled MSIF.

119


inside a horizontal furnace in such a way such that the gold-filled section is at one end

of the heat zone. The temperature profile of the furnace must be checked to determine

Molten goldSilicajacket tube

Heat zoneof the furnace

cane

vaccum

Glue

Figure 6.16.: Schematic diagram showing the filling of a cane with gold using vacuum.

the uniformity and length of the heat zone. The temperature is then ramped up, to

achieve a temperature just above the melting point of gold (1350K). After approximately

5 minutes, the cane is pushed inside the jacket (1-2mm) so that the end face of the cane

is in good contact with molten gold. A vacuum is then applied into the hole of the cane.

Due to a pressure difference of 1 bar, the molten gold is sucked into the empty hole of

the cane. The vacuum is applied for 10 minutes; after that the cane is pulled out from

the jacket tube and the furnace is switched-off. Using this technique, a hole with an ID

of 90µm can be easily filled ≈ 70cm. The continuous length of the gold column in the

filled cane is approximately ≈ 40cm.

The minimum hole diameter which can be filled with molten gold using vacuum was

found to be ≈ 18µm. (section 5.3 and ). The pressure required to fill a 90µm hole with

the molten gold is ≈ 0.36bar. It is well below the pressure difference due to vacuum

(1bar).

Step 2: In the second step, the outer surface of filled cane is cleaned and the cane is

placed inside a suitable jacket tube.

Step 3: The jacketed cane is then fixed on the preform chuck of the fiber tower. The

fiber drawing process is similar to the one explained in section 6.1.2. The main difference

between drawing an empty MSIF and a filled MSIF cane are the drawing parameters.

To avoid any diffusion of gold into the glass, the filled MSIF is drawn at relatively low

temperatures (≈1921°C) and relatively high drawing speed (≈ 100m/min).

Due to the surface tension of gold at these temperatures, molten gold inside the fiber

tends to form gold spheres. The thermal expansion coefficient of gold is larger than

silica (section 5), which results in the formation of gold columns of certain lengths. The

120

6.4. Characterization techniques

length of the continuous gold column depends on the diameter of the hole and The

smaller diameter results in to a smaller length of the columns. Higher drawing speed (≈100m/min) and low drawing temperature allows gold to spend less time in the furnace,

which helps to reduce the possibility of diffusion of gold in silica Figure 6.17 shows the

100nm

Figure 6.17.: SEM image of gold-filled capillary with only one hole fabricated using the directdrawing method. For SEM image of Au-filled MSIF please check figure 8.1.

SEM image of a gold-filled capillary, which was fabricated using the direct drawing filling

technique [74].

6.4. Characterization techniques

One of the most important feature of filled structures is the continuity of the wire (Au

or Ge) along the length of PCF/MSIF. The most efficient way to measure the continuity

of wire is to measure the conductivity of the sample. The structural properties of filled

structures can be investigated by micro Raman measurements.

6.4.1. Conductivity measurement

The conductivity measurements are useful in determining the continuity of germanium

or gold wire inside the PCF/MSIF. For germanium, these measurements also provide

information about the resistivity of filled germanium. This resistivity value also reveals

some information about the level of crystalline germanium inside the fiber.

121


These measurements require the connecting of the two end faces of filled structure with

the outer circuit. A voltage is then applied using a voltage source to the outer circuit

and resistance/conductivity of the sample is measured. Figure 6.18 shows the schematic

design of the setup which is used to measure the conductivity of filled germanium/gold

-filled single hole capillary.

For germanium, a commercially available high resistance meter HP A4339 is used to

measure the resistance. This device is specially designed to measure high resistances of

the order of 1011 Ω. As an output, this device can provide the value of current passing

through the sample and the resistance of the sample for a given applied voltage. First,

the intrinsic resistance of the sample is measured without any light at room temperature.

Connectionwire

Connectionwire

HP 4300Resistivity

meter

Gallium drops

Figure 6.18.: Schematic diagram of the experimental set up used to measure the conductivityof germanium/gold -filled PCF, MSIF or a single hole capillary.

Gallium6 is used as the contacting material between the germanium wire and outer cir-

cuit. As a first step of the contacting process, two separate drops of gallium are placed

on a fused silica slide. This base silica slide is placed on an adjustable 3-dimensional

translational stage. The drops are placed at a spacing which is equal to the length of the

sample under investigation. The diameter of the each gallium drop is ≈ 5-6mm. The

two end faces of the sample are then very carefully inserted into the gallium drops. It is

important to place the two end faces of the sample inside the gallium drop in such a way

that both the end faces are completely inserted into the gallium spheres. Another im-

portant point is that the filled sample should not touch the silica base plate on which the

6Gallium is a metal which is in liquid phase at room temperature.

122

6.5. Optical measurement techniques

gallium drops are placed. Therefore, any measurable current in the circuit(conductivity)

is only due to the germanium wire inside the PCF/MSIF. Now, two gold wires are also

carefully inserted into the gallium drops. The opposite ends of these gold wires were

connected with the connection wires of the resistance meter using standard electrical

clamps.

Once the connections are made, the voltage is applied and the value of resistance can be

obtained from the resistivity meter. The resistivity of the material is then calculated as

ρ = Rπr2

l(6.3)

here ρ is the resistivity of the filled material, r is the radius of wire, l is the length of the

wire and R is resistance of the germanium wire.

6.4.2. Raman spectroscopy

Structural properties play a very important role in describing optical and electrical pro-

perties of germanium. Micro Raman measurements are used to investigate the crystalline

behavior of the filled sample. A micro-Raman spectrometer7 is used to characterize Ge

wires. Light from a HeNe laser (wavelength 632.8 nm) is focused through the cladding

on the wire with a 100X objective and the back-scattered light is then coupled into the

spectrometer. The Raman measurements are made at several positions along the filled

structure to check the uniformity of germanium/silicon along its length.


Optical measurements of filled structures are very useful in investigating the interaction

of propagating light in the core of a PCF/MSIF with gold or germanium. One basic way

to perform optical measurements is to check the transmission spectrum of the filled fiber

and to compare it with the unfilled fiber.

7(Jobin Yvon LabRAM HR 800).

123


6.5.1. Optical transmission measurement

Optical transmission spectra of the core of PCF/MSIF are measured and compared with

the unfilled PCF/MSIF. This spectrum is measured using a ESM PCF based broadband

light source (400-1600nm)) referred to as supercontinuum (SC) source. The process of

SC generation has been explained by several researchers, e.g. Ranka et al. [75]

Micro chip laser1064nm

Polarizer

Half waveplate

Optical SpectrumAnalyser

(OSA)

Iris

Objective

Alignmentmirror

ESM PCFfor SC

generation

Beam path

Glass slide with indexmatching liquid

Sample

Ge filledsection

Half waveplate

Polarizer

Figure 6.19.: Schematic diagram of the experimental set up used to investigate the transmissionspectrum of germanium-filled samples.

A schematic image of the experimental setup to measure the transmission spectrum is

shown in figure 6.19. In this experiment the SC source consists of a pump source, a

sealed microchip IR laser with a pump wavelength of 1064nm and a repetition rate of 25

kHz. This laser radiation is coupled into a silica ESM PCF (l = 20m)for supercontinuum

generation.

The light from the output end of this fiber is collimated using an achromatic objective lens

(40X). The collimated beam is reflected through two alignment mirrors and coupled into

the sample using an achromatic objective lens (numerical aperture (NA) of the objective

lens depends on the fiber core size). To control the polarization state of the input light,

a half-wave plate and a polarizer are placed in front of the incoupling objective. To

achieve the maximum coupling efficiency, the sample is clamped on to the fiber holder,

which is fixed on a precision 3-dimensional translational stage. If the sample is long(l

>5cm), then a second stage (precision 3-dimensional translational stage) is used to fix

the output end of the fiber using a fiber holder. The light from the output end of the

124


fiber is collected and collimated using an achromatic objective lens of correct NA. This

collimated light coming out from the objective lens is coupled into the optical spectrum

analyzer (OSA)8 using two alignment mirrors. To check the polarization of the output

beam, a combination of polarizer and half-wave plate are placed in the beam path before

the OSA. A pinhole iris is also placed just before the OSA which helps to block stray

light from the PCF cladding.

Sometimes, index matching liquid is also used to filter out the cladding light, which helps

to reduce the effect of cladding light in the transmitted spectrum of the glass-core.

The optimization of the output light is achieved by first coupling the light from the core

of the fiber sample in the OSA at a specific wavelength. The maximum power at this

wavelength is then obtained by slightly adjusting the mirror and position of the stages

and measuring the output power. Usually, the output spectrum is measured for different

optimization wavelengths.

The general procedure of investigating the transmission characteristics of the sample is

to measure the spectrum for filled and unfilled fibers and normalize the filled spectrum.

This gives the normalized transmission spectrum of the filled fiber.

The loss(dB/cm) spectra is determined using the cut-back technique. In the cut-back

method, the transmission spectrum of the Au-filled MSIF is obtained by step by step

reducing the length of the filled MISF. At each length of the sample, for a fixed wave-

length, transmission is plotted against the length (in cm) of the sample. The slope of

this data is then calculated by linear fitting the data points of each wavelength. This

calculated slope is the loss of the sample in dB/cm at that particular wavelength. This

process is then repeated for each wavelength to calculate the loss spectra.

6.5.2. Photoconductivity measurement

As discussed in section 3.4.3, an absorbed photon (having sufficient amount of energy) in

germanium leads to the generation of charge carriers in the form of an electron and a hole.

Thus, these germanium-filled fiber structures can be used as an in-fiber photodetector.

These carriers can be excited in two experimental configurations: (i) by side pumping of

8A grating based optical spectrum analyzer (OSA) (from commercial supplier Fuchukawa) is used inthis experiment. It measures the transmitted power at each wavelength, by diffracting the beam intoits spectral components using a grating. The spectrum of the power can then be plotted on dBmscale or on linear scale.

125


light and (ii) in transmission mode. The experimental setup for these two configurations

is discussed in this section.

6.5.2.1. Side pumping

One way to modulate the resistivity of germanium-filled PCF/MSIF is to focus the light

on the fiber from the side. This technique can also be used to measure the change in

resistivity in a single hole capillary (the presence of core is not necessary for this mode

of measurement) filled with germanium. A Verdi laser, operating at 532 nm is used as

the light source.

Optional inputpower meter

Connectionwire

Connectionwire

HP 4300Resistivity

meter

High powerVerdi laser(532 nm)

e h

Gallium drops

Core

Ge-filledhole

+ -

Elliptical focus spot

Figure 6.20.: Schematic diagram of the experimental setup for the measurement of photocurrentfrom a Ge-filled MSIF using side pumping with verdi laser. The operating wavelength of theVerdi laser is at 532nm.

Figure 6.20 shows the schematic design of the experimental setup used to measure the

photoconductive effect in the side pumping mode. The wire in the sample is connected

to the outer circuit using gallium drops (section 6.4.1).

As a test experiment, the resistance of the empty capillary is first measured to obtain

the reference of the setup. As expected, the empty PCF/MSIF does not show any sign

of conductivity. In the second measurement step, the resistance of the germanium-filled

sample is measured. Now, in order to measure the photoconductivity of the sample,

the laser light reflected through two alignment mirrors is focused on the wire using a

126


cylindrical lens of focus length ≈ 15mm (as shown schematically in figure 6.20). Due to

the concave curvature of the silica surface, a large fraction of incident light is scattered

and reflected, which can be potentially minimized by using index matching liquid at the

surface.

An optional input power meter is also used to measure the incident power on the fiber

surface. The variation of resistance/current with respect to incident power is measured

for various applied voltages.

6.5.2.2. Transmission mode

An in-fiber photodetector can be realized by measuring the photoconductivity, while

the light is guided through the core of the fiber. In this configuration, light gently

interacts with the germanium wire and generates charge carriers . This device can be

fabricated only for a germanium- filled PCF/MSIF as it requires a light guiding core.

Figure 6.21 shows the schematic diagram of the experimental setup used to measure the

photoconductivity in transmission mode.

An electrical contact between the germanium wire of the sample and the outer circuit

is a bigger challenge in this type of measurement. In this case the electrical contacts

cannot be made using gallium drops, as light has to be coupled into the fiber core. It

means that the electrical contacts must be made in such a way that both transmission

measurements and electrical resistivity measurements can be carried out simultaneously.

In these measurements, electrical contacts are made by sputtering a thin layer (30nm) of

indium tin oxide (ITO) on both end faces of the fiber. ITO is one of the most widely used

transparent conducting oxides because of its two main properties- namely the electrical

conductivity and the optical transparency.((In2O3) and tin oxide (SnO2), typically 90%

In2O3, 10% SnO2 by weight). It is transparent and colorless in thin layers, while in bulk

form, it is yellowish grey. In the infrared region of the spectrum it behaves as metal-like

mirror. Like other transparent conducting oxides, a compromise has to be reached for

the layer thickness. A thicker layer increases the conductivity of ITO, but decreases its

transparency.

The ITO is sputtered on the sample in such a way that 6mm length of the fiber in the

middle of the sample is completely masked from ITO. It allows sputtering on both end

faces of the fiber, while keeping the middle region un-sputtered. The most appropriate

127


thickness was found to be ≈ 30nm. The sputtered sample is fixed on a custom made

fiber holder, made of hard Teflon (instead of a metal fiber holder), which helps to avoid

any electrical contact of sample with the metallic fiber holder or translation stages. The

width of the Teflon holder is approximately 2mm. The sample fiber is placed on this

holder in such a way that only the middle un-sputtered region is in contact with the

holder. This fiber holder is fixed on a 3-dimensional translational stage.

The two sputtered end faces are then connected with two separate gold wires (diameter

0.1mm to 0.5mm) using a layer conducting glue. As only the middle 6mm region is

masked, a layer of ITO is also sputtered on the cylindrical surfaces of both ends of the

sample; it is easy to connect the wire with these cylindrical surfaces. The opposite ends

of these two thin Au wires are then connected to the connection wires of HP resistivity

meter by using the standard electrical clamp clips. The sample is now electrically con-

nected to the outer circuit. The experimental setup shown in figure 6.21 is similar to the

PolarizerHalf wave

plate

Alignmentmirror

Beam path

HP 4300Resistivity

meter

Connectionwires

Outputpowermeter

Optionalinput power

meter

ObjectiveSample

Germanium filledmodified step index fiber with

ITO sputtered end faces

Iris

Half waveplate Polarizer

Multi-colorHe-Ne laser

(+)

(-)

Figure 6.21.: Schematic diagram of the experimental setup for the measurement of photocon-ductivity from a Ge-filled MSIF using a multi-color He-Ne laser. The operating wavelength usedfor the experiment is 543nm.

transmission measurement setup explained in section 6.5.1. In this case, the light source

is a tunable multi-color He-Ne (633, 612, 604, 594, and 543 nm). The choice of operating

wavelength for the experimental setup (out of the five available colors) depends on the

transmission spectrum of the Ge-filled MSIF. An optional power meter can also be used

128

6.6. Simulation techniques

before the incoupling objective lens to measure the power which is coupled into the fiber.

Initially, the optical transmission spectrum of the sample must be measured to find the

suitable wavelength in order to observe photoconductivity. The polarization properties

of the sample are also investigated to understand the effect of polarization of light on the

photoconductivity.

6.6. Simulation techniques

Theoretical analysis plays a very important role in understanding the fundamental physics

and analyzing the experimental results.

6.6.1. Finite element method: JCMWave

Finite element method based commercial software JCMWave is used to theoretically

investigate the the filled structures. Both germanium and gold have its intrinsic absorp-

tion loss, which means that full dielectric function must be included in the calculations

to simulate the loss of full structure.

129

Chapter 7Results: germanium-filled structures

7.1. Material characterization of Ge-filled structures

Before measuring the optical properties, the physical characterization of the germanium-

filled samples are performed by using the characterization techniques discussed in section

6.4.

7.1.1. Conductivity measurements

The electrical conductivity of germanium-filled samples is measured based on the tech-

nique explained in section 6.4. Experimentally measured electrical resistivities of three

different types of Ge-filled structures are listed in table 7.1.

The resistivity of undoped germanium is reported as ≈ 47 Ω-cm at room temperature

[76] [77] [44]. The measured resistivity of each filled sample is slightly higher than the

reported one, because the germanium wires inside the samples are not single crystalline,

Ge-wire diameter (µm) length (cm) resistivity(Ω-cm)

Ge-filled capillary 1 1.8 45

MSIF 1.1 1.5 50

PM-PCF (figure 6.14) 0.6 2.2 49

Table 7.1.: Resistivity of three different germanium-filled samples.

131

7. Results: germanium-filled structures

but rather polycrystalline. The grain boundaries and surface states work as trap centers

or energy barriers for the charge carriers, which reduces the effective motion of the charge

carriers. Despite this effect, the value of the measured conductivity is quite close to the

reported ones.

7.1.2. Micro-Raman measurements

The micro-Raman measurements of the filled samples are performed using the technique

discussed in section 6.4.2. For silicon and germanium, the spectra of different samples

are measured at different positions along the axis of the fiber. Figure 7.1 shows one of

the resulting spectra for silicon and germanium respectively. The main features of the

measured spectra are as follows:

Germanium : The micro-Raman spectrum of germanium shown in figure 7.1(b) is

measured for a Ge-filled capillary of diameter 1.9µm. The spectrum shows a symmetric

peak at around 298cm−1. This peak corresponds to the transverse optical (TO) Raman-

active mode of the germanium crystal.1 For a bulk single crystal germanium, the TO

Raman-active mode peak was observed at 300cm−1 [78] [79].

The measured peak is shifted to lower frequency by ≈ 2cm−1. The Raman spectrum for

an amorphous germanium shows a broad shoulder at low frequencies ≈ 271cm−1, which

indicates that the germanium inside the filled capillary is highly crystalline. Another

important feature of a Raman spectrum is the full width at half maximum (FWHM) of

the peak. The measured TO-peak in this spectrum has a width of ≈ 3.6cm−1, which

is slightly wider in comparison to that of single crystal germanium (2.4cm−1) and much

narrower than that of amorphous germanium (>50cm−1) [78]. This comparative analysis

of width of TO-peak also supports the argument that the germanium in these filled

samples is highly crystalline (close to single crystal).

Silicon : Figure 7.1(a) also shows the micro-Raman spectrum of a silicon sample for a

silicon-filled capillary of wire diameter ≈ 90µm. The spectrum shows a symmetric peak

at ≈ 519.81cm−1 with a FWHM of ≈ 3.921cm−1. This peak at 519.81cm−1 corresponds

to the scattering of a first-order optical phonon at the center of the Brillouin zone in

crystalline-Si. For a bulk single crystal silicon, Raman-active peak was observed at ≈521cm−1 with a FWHM ≈ 4.6cm−1 [78] [79]. The small shift of the Raman-peak to lower

1The TO Raman-active modes are caused by the vibrations of positive and negative ions against eachother, creating a time-varying electrical dipole moment in the transverse direction.

132

7.2. Optical characterization of Ge-filled structures

Raman shift (cm )-1

3.9216cm-1

500 510 520 530 5400.0

0.2

0.4

0.6

0.8

1.0

Norm

aliz

ed inte

nsity (

a.u

.)

519.81cm-1

Silicon

291 294 297 300 303

0.2

0.4

0.6

0.8

1.0

3.6cm-1

Raman shift (cm )-1

297.6 cm-1

Germanium(a) (b)

300cm-1

521cm-1

Figure 7.1.: Micro-Raman spectrum (a) silicon wire of diameter 90µm (b) germanium wire ofdiameter 1.6µm, measured through the side of the Ge/Si filled capillary. The value of reportedRaman peak for bulk single crystalline Si/Ge is also mentioned in respective plots.

frequency can be attributed to its polycrystalline nature of the material and also the

fact that silicon inside the capillary is under under stress [79]. The Raman spectra of an

amorphous silicon shows a broad shoulder at lower frequencies, which means that silicon

is in polycrystalline state inside the capillary .


The optical characterization of fabricated Ge-filled samples are performed by measuring

the transmission spectra. The measurement technique and experimental setup used to

measure the transmission spectrum of a sample are explained in section 6.5.1.

7.2.1. Completely filled ESM PCF

In the first attempt, an ESM PCF is filled with germanium using the pressure cell filling

technique explained in section 6.3.1. The ESM PCF is chosen because it supports only

the fundamental mode at all wavelengths, which is very helpful for investigating the light-

semiconductor interaction in a wide range of wavelengths. Figure 7.2(a) shows the SEM

image of the filled structure. The diameter (d) of the filled holes is ≈ 1.0 µm and hole-to-

hole spacing, i.e. pitch (Λ) is ≈ 2.9 µm. The total sample length used in this measurement

is ≈ 50cm in which one end is filled with germanium with a of length ≈ 0.5mm. The

133


presence of an unfilled part of ESM PCF before the filled section helps to have well-defined

launch conditions for light at all wavelengths. All the holes of the first ring are filled with

germanium, while some of the other holes (particularly in the outer rings) remain unfilled.

The interaction (overlap) of the glass-core mode is maximum with the holes in the first

ring. Hence, the transmission properties of this structure would be more or less similar

to a completely filled structure. Figure 7.2(b) shows the transmission spectrum of the

10 mμ

(a)

600 800 1000 1200 1400

-60

-50

-40

-30

Norm

aliz

ed tra

nsm

issio

n (

dB

)

Wavelength (nm)

(b)

Figure 7.2.: (a) SEM image of first ESM fiber (pitch: 2.9 µm, hole diameter: 1.0 µm) filledwith germanium. (b) Normalized transmission spectrum of the sample.

structure (7.2(a)). It is clear from figure 7.2(b) that the transmission drops on certain

wavelengths, forming transmission dips. These transmission dips arise from a coupling

of light from the glass-core mode to the individual modes of germanium wires as well as

with the super-cladding mode of the Ge-filled cladding. The high magnitude of loss for

the dip around ≈ 1000nm and a wide transmission dip over a broad wavelength range

from ≈ 1200nm to ≈ 1450nm makes it difficult to understand the nature and resolution

of dips due to high loss. To overcome this problem and to understand the physics behind

this interaction only one hole in the vicinity of the glass-core is filled with germanium.

7.2.2. Single hole filled ESM PCF

The high loss of completely filled structures shown in section 7.2.1 can be controlled by

filling just one single hole adjacent to the core. These single wire structures not only

results into well resolved dips in the transmission spectrum, but also helps to understand

the coupling between the modes of glass-core and germanium-wire.

134


These single hole-filled samples are fabricated by first selectively opening only one hole on

one of the end faces using the techniques explained in section 6.2.3, and then filling them

with germanium, using the techniques explained in section 6.3. Figure 7.3 shows the SEM

image of an ESM PCF (diameter (d) = 1.7µm, hole spacing (Λ) = 2.90µm), in which a

single hole adjacent to the core is filled with germanium. The overall length of the sample

is ≈ 50cm. It consists of a large unfilled section ≈ 49cm followed by a single Ge-wire

filled section of length ≈ 1cm. The length of the filled section is reduced to a length that

provides a suitable transmission spectrum with optically resolved dips. The presence of

x

y

2 µm2 µm

(a) (b)

Figure 7.3.: (a) SEM image of ESM PCF with a single hole adjacent to the fiber core filledwith germanium. (b) Enlarged view of the glass-core and single wire of germanium-filled inESM PCF. The axis indicates the direction of polarization of the incoupled light.

only one guided core-mode and unfilled section at the incoupling side of sample helps

to have well-defined and reproducible launch conditions at all wavelengths. There is a

possibility that higher order modes might be excited by imperfections at the filled-unfilled

transition. The transmission measurements of these structures were performed using the

optical measurement technique discussed in section 6.5.1. As only one hole adjacent to

the core is filled, the six-fold symmetry of the ESM fiber core is broken. Thus, it is

important to understand the effect of this modification on the polarization properties of

the filled fiber. For this structure, a combination of a polarizer and a half-wave plate

is placed in the path of light which is coupled out from the sample. The x- and y-

coordinate axis depicted in figure 7.3 correspond to the x- and y- polarizations of light

in the experiments.

Transmission in the range from 500nm to 1050nm: The experimental transmission

spectra and corresponding numerical simulations (using commercial software JCMWave)

for x- and y- polarizations2 are shown in figure 7.4(a) and (b), respectively. The length

2The axis of polarization are marked in figure 7.3.

135


wavelength (nm)

600 700 800 900 1000

Ty

Tx

500 600 700 800 900 1000

-24

-18

-12

-6

0

Tx

Tra

nsm

issio

n (

dB

)

Ty

wavelength (nm)

(a) (b)

Figure 7.4.: Experimental (a) and simulated (b) transmission spectra for x- and y- polarization(Tx and Ty respectively) for structure shown in figure 7.3. The inset of the left figure showsthe CCD image of the transmitted mode pattern for y-polarization. For simulations, the funda-mental mode of the glass core was selected for numerical investigations. The inset of the rightfigure shows the selected mode (axial component of the Poynting vector for y-polarization) forsimulations at a wavelength of 550nm.

of the germanium wire is 0.8mm. There is a good agreement between the experiment

and simulations at all wavelengths, which allows us to infer that the polarization state is

preserved when light crosses into the Ge-filled section. Both experiment and simulation

show an overall drop in transmission at longer wavelengths, caused by an increasing

field overlap between the glass-core mode and the Ge-wire (to be explained later in this

section). Although the intrinsic absorption of germanium drops at longer wavelengths

in these structures, the increase of field overlap dominates over the drop in absorption

at longer wavelengths. Experiments and theory follow the same trends with an offset

of roughly 3dB. This offset can be explained using two arguments: First, the excitation

of higher order modes at the interface between the filled and unfilled section increases

the measured loss (For simulations, only the loss of the fundamental mode is plotted).

Second, the dielectric function of germanium used in the simulations applies for the single

crystalline material, whereas the Ge wire is polycrystalline. It increases the scattering

and therefore also increases the loss. In both spectra, weak dips appeared in the range

from 900nm to 1000nm, which are the result of light coupling from the glass-core mode

to resonances on the Ge-wire. More pronounced dips can be observed for wavelengths

longer than 1000nm (will be discussed later in this section). Near field images at the end

faces confirmed that only the fundamental mode of the glass-core is present in all cases

(inset in figure 7.4.)

136


Another important observation of the transmission spectrum for this type of structures

is the study of polarization properties. Due to the presence of a single Ge-wire adjacent

to the core, the degeneracy of the fundamental mode of the ESM PCF is broken. Figure

7.5 shows the polarization behavior of the transmission spectrum.

500 600 700 800 900 1000-30

-20

-10

0

Tra

nsm

issio

n r

atio (

dB

)

Wavelength (nm)

0 45 90 135 180

-40

-30

-20

? = 950nm

Analyzer angle

T(d

B)

y x y

500 600 700 800 900 1000-50

-40

-30

-20

-10

0

Tra

nsm

issio

n (

dB

)

Wavelength (nm)

28°

0°

48°

68°

78°

88°

90°

(a) (b)

Figure 7.5.: (a) Transmission as a function of wavelength for various positions of the outputpolarizer. (b) Ratio (in dB) between the minimum and maximum transmission as a function ofwavelength. The inset of right figure shows the transmission versus angle of the output polarizerat a fixed wavelength ≈ 950nm. These measurements are obtained for the structure shown infigure 7.3 with 1.7mm length of Ge-wire.

It is clear from figure 7.5(a), that for each wavelength, the transmission gradually de-

creases as the polarization axis changes from y-polarization (00) to x-polarization (90o).

The overall decay in transmission at longer wavelengths for all polarizer states occurs

due to increased field overlap at longer wavelengths. Figure 7.5(b) shows the variation

of Tx/ Ty (in dB) as a function of wavelength. It is clear from figure 7.5 that minimum

transmission is obtained for x-polarization. The ratio of transmission also increases at

longer wavelengths, reaching a maximum of 28dB at around 850nm. This high value

can be explained by the fact that the power absorbed at the surface of the conductor

is proportional to the tangential magnetic field inside the conductor, i.e. H parallel to

the surface [28]. For x-polarization, the tangential magnetic field in the wire is much

stronger than in the orthogonal case. Since germanium has finite conductivity at the

optical frequencies, higher losses are expected for x-polarized light. Apart from some fea-

tures appearing for wavelengths larger than 900nm, the transmission is quite flat without

any pronounced dips. This type of structure could be used as an in-fiber polarizer.

Transmission in the range from 1050nm to 1500nm Experimentally measured and

simulated loss spectra of the structure (figure 7.3) for x- and y- polarizations are shown

137


in Figure 7.6, in the wavelength range from 1050nm to 1500nm. Each experimental

spectrum is normalized to the corresponding unfilled fiber. The spectra shown in 7.6

Wavelength (nm)

1100 1200 1300 1400 15005

10

15

20

25

30

EH16 TE05

EH24

EH15

Experiment

Theory Y-pol

TE04

Loss (

dB

)

1100 1200 1300 1400 1500

20

40

60

80

100

HE24

HE15

TM05

HE16

Experiment

Theory

X-pol

TM04

Wavelength (nm)

(a) (b)

Figure 7.6.: Loss spectra of the structure shown in figure 7.3 for Ge-wire of the length of 0.8mm.(a) Experimental (black curve) and simulated (right curve) spectra of the glass-core mode of thestructure for x-polarization. (b) Experimental (black curve) and simulated (right curve) spectraof the glass-core mode of the structure for y-polarization. The labels on the peaks of refer tothe resonances on the Ge-wire that phase match to the glass-core mode. Three encircled pointscorrespond to the modes whose Poynting vector distribution is shown in figure 7.7.

display multiple pronounced peaks, which are caused by the coupling of the glass-core

mode to successive resonances on the Ge wire. These peaks occur for wavelengths at

which the dispersion curves for the wire-resonances and PCF glass-core mode anti-cross,

causing the light to couple strongly with the resonances on Ge-wire. This coupling

enhances the losses at these resonance wavelengths. The modes guided in the glass-

core have effective phase indices that lie below that of the silica and above that of the

fundamental space-filling mode (FSM) in the PCF cladding, which means that the anti-

crossings occur at Mie resonances of the Ge wire.

The experimental positions of the transmission dips agree with finite-element simulations

to within 1%. This close agreement between the experiment and simulation is achieved

despite the fact that the idealized structure has been used in the simulations and that

there are uncertainties in the value of the Ge dielectric function in the infrared region

(especially the imaginary part). The dips for two orthogonal polarizations are not always

located at the same position, which indicates that the mode coupling between the Ge-

wire mode and glass-core mode depends on the polarization. This is especially clear for

the loss peaks marked TM05 and TE05 in figure 7.6. For x-polarization, this peak appears

at around 1206nm, whereas for y-polarization it is at 1182nm.

138


The mode orders mentioned above each of the loss peak in figure 7.6 could be identified,

by comparing the axial Poynting vector distributions inside the Ge-wire at each anti-

crossing resonance wavelength (calculated using the finite-element code) with those at

resonances in an isolated Ge-wire embedded in silica. The calculations were performed

for the three resonance wavelengths marked by small blue circles in figure 7.6, i.e. 1110nm

for y-polarization marked as resonance order EH16; 1201nm for x- and y-polarizations

marked as resonance order TM05; and 1397nm for x-polarization. The top three mode

1201nm, X-pol., TM05 1397nm, X-pol., HE241110nm, Y-pol., EH16

No3a7M No4a9M

(a)

(d)

(b)

(e)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

x/R

y/R

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

0.00

0.25

0.50

1.00

0.75

(c)

(f)

-1.0 -0.5 0.0 0.5 1.0

Figure 7.7.: Axial Poynting vector distributions in the vicinity of Ge-wire (top row form a-c)in a filled ESM PCF calculated by using finite-element simulations. The bottom row (from d-f)shows the axial Poynting vector for a Ge wire embedded in silica capillary, calculated by directlysolving Maxwell’s equations for the Mie resonances on the wire. The wire radius in both casesis R = 0.85µm.

patterns shown in figure 7.7 (corresponding to these three wavelengths) are obtained

from finite element simulations. The lower three modes are obtained using analytical

calculations for a Ge-wire in a silica capillary (wire diameter 0.85µm) at these resonance

wavelengths. By comparing the mode patterns of FEM simulations with the single wire

calculation, the mode order can be determined. It should be noted that for single wire

calculations, the fields outside the wire have been removed to highlight the internal field

patterns of the wire. At longer wavelengths, when coupling to the Mie resonances of the

Ge-wire is stronger, the mode patterns differ more noticeably, although it is still possible

139


to identify the mode order accurately using this method.

From figure 7.6(b) it is clearly evident that for y-polarization, the magnitude of the loss

peaks in experiment matches quite well with the corresponding loss peaks obtained from

simulations. For x-polarization, the magnitude of loss at the peaks was very high and

could not be measured accurately for λ0 > 1200nm, resulting in a significant disparity

between experiment and theory, particularly for HE15. This discrepancy can be explained

by taking into account that the filled ESM PCF structure (figure 7.3) supports more than

one modes in the glass-core. Figure 7.8 shows the simulated loss (in dB/cm) of two glass-

Fundamentalmode (X pol)

Fundamentalmodel (Y pol)

First order (X pol)

First ordermode(Y pol)

500 600 700 800 9000

5

10

15

20

25

Lo

ss (

dB

)

Wavelength (nm)

Figure 7.8.: Simulated loss (in dB) spectra for fundamental and first higher order mode sup-ported by the glass-core in x- and y- polarizations of the structure shown in figure 7.3.

core modes (fundamental mode and first higher order mode) for x- and y- polarizations.

It is clearly visible from figure 7.8 that for y-polarization, the fundamental mode has a

higher loss than the first order mode, but for x-polarization, the first order mode has

a much lower loss than the fundamental mode. This low loss behavior of first order

mode in x-polarization is due to a node in the electric field distribution along the central

mirror-plane of the structure. Such higher-order glass-core modes, if excited even slightly,

can significantly increase the transmission, reducing the measured loss at the peaks.

Despite this, the relative magnitude between the loss peaks, and the tendency for higher

resonant losses at longer wavelengths, are predicted correctly. Between the peaks, the

slightly higher experimental loss value can again be ascribed to modal mismatch at the

filled-unfilled interface, which is an effect that cannot be removed from the experimental

140


data.

It is also noticeable that for y-polarization, the loss peak near 1450 nm is not clearly

resolved in the experiments (as shown in figure 7.6(b)). As discussed earlier, this can

be explained due to the fact that light guided in the higher-order glass-core modes con-

tributes significantly to the overall measured transmission at longer wavelengths, masking

the loss peak of the fundamental mode. For the calculated TE04 and TM04 modes, the

loss peak shows a spectral splitting that is weakly present only in the experimental x-

polarized spectrum.

Effect of Ge-wire diameter and pitch of ESM PCF on loss peaks:

The dimensions of the ESM PCF plays an important role in determining the position of

the loss peak and magnitude of the loss. As discussed earlier in this section, the origin of

the loss peaks are the coupling of the glass-core mode to different resonances on the Ge

wire. The results indicates that the position of the loss peak is determined by the position

of the anti-crossing between the mode of the wire and the glass-core mode. Therefore,

the position of the loss peak is a function of the wire diameter. Figure 7.9 shows the

1640 1660 1680 1700

1080

1140

1200

1260

1320

1380

Experiment

Resonance p

eaks

Ypol (n

m)

Wire diameter (nm)

E = 0.68H15

T = 0.58E05

E = 0.59H16

E = 0.74H24

Figure 7.9.: Variation of resonance wavelengths (loss peaks shown in figure 7.6(b)) with respectto wire diameter.

variation of loss peaks with wire diameter for y-polarization. It clearly indicates that

each resonance peak shift towards the longer wavelength as the wire diameter increases

and the slope for each peak is different. The slope of the peaks in y- polarization are

listed in table 7.2.

141


Loss Peak (mode order) slopes

EH24 0.74

EH15 0.68

TE05 0.58

EH16 0.59

Table 7.2.: Slope of the different loss peaks in y-polarization for a variation in Ge-wire diameter.

Figure 7.9 also shows the position of experimentally obtained loss peaks (y-polarization)

for the structure shown in figure 7.3.

Another important feature of the ESM PCF is the pitch which determines the spacing

between the holes. The major effect of the increase in pitch is that the effective mag-

nitude of the overlap between the Ge-wire mode and glass-core mode decreases, which

consequently decreases the magnitude of the loss peaks. By carefully designing a hybrid

structure, the position and magnitude of these peaks can be effectively controlled.

Thin layer model

As pointed out in the previous section, the modes guided in the glass core have an

effective index that lies below that of silica, indicating that the Ge wire, which sits inside

the silica guiding core of the PCF, cannot support properly bound modes, but rather

will exhibit Mie-like resonances for certain combinations of wavelength and axial phase

index. When the glass-core mode passes through a wavelength region in which it phase-

matches to a resonance on the wire, its attenuation rises and its phase index undergoes

a local distortion. To understand the physics of this interaction quantitatively, a simple

thin-film waveguide model is investigated. The thin film waveguide consists of a layer of

silica (4µm wide) and a layer of Ge (1µm wide), which is sandwiched between a cladding

material of refractive index 1.35. The full dispersion of Ge and silica is included in these

calculations and the cladding material is assumed to be without dispersion.

Figure 7.10 shows the results of the analysis for this simple thin layered structure. The

main features of the loss spectrum (dB/cm) shown in the figure 7.10(a) reveals that

these features (loss peaks) are quantitatively similar to those seen in a Ge-filled PCF

structure (figure 7.3), i.e. the appearance of loss peaks at wavelengths beyond 900nm,

a much higher loss for the TM mode, and a gradual reduction in average loss at shorter

wavelengths. The origin of these peaks can be better understood by plotting the modulus

142


600 800 1000 1200 1400

1

10

100

TE

Loss (

dB

/mm

)Wavelength (nm)

TMA

B

C

D

-1

0

1

2

3

4

5

6

A B C Dcladding

silica

Ge

cladding

Tra

nsvers

e c

oord

inate

z (

µm

)

|E +E |x z

2 |E |y

2

(a)(b)

Figure 7.10.: (a) The loss spectra (dB/cm) of the fundamental TE and TM guided modes insilica layer (4µm wide) and a layer of Ge (1µm wide), sandwiched between cladding material ofrefractive index 1.35. (b) Modulus square of the electric fields for four modes marked as A, B,C and D in left figure.

of the electric field at the position marked as A, B, C and D in the loss spectra of figure

7.10. The presence of a resonance in the Ge layer causes the loss to peak strongly at

resonance positions (B and D). In between resonances, the loss falls below its average

value.

This simple model confirms that the TM mode (closely related to the x-polarized mode

in the PCF) experiences much larger loss, that the loss peaks occur when a resonance is

strongly excited in the Ge layer, and that loss peaks fade away at wavelengths below 900

nm (figure 7.10(a)). The distributions of electric field in figure (7.10(b)) illustrate that

the loss is highest when a resonance exists in the Ge layer (B and D), while being much

lower at anti-resonances (A and C).

The loss spectra for the thin layer model (figure 7.10) and loss spectra for the Ge-filled

ESM PCF (figure 7.4 and 7.6) conclude that for wavelengths shorter than 800nm, there

is very weak coupling between the silica core and Ge-modes. This weak coupling can

be attributed to the fact that the high loss of germanium in this wavelength range does

not allow the mode to form in the Ge-wire due to extremely small penetration depth.

In a ray picture, a mode is formed by a series of the total internal reflections from the

surface of Ge-wire but if the penetration depth is so small that the light ray get absorbed

before a complete cycle of TIR, a mode cannot be formed. As discussed in section 5.2.1,

the imaginary part of the dielectric function increases with decrease in wavelength in

143


the range from 400nm to 1600nm. Using FEM simulations, it can be explained that

this high absorption is indeed the reason for absence of loss peaks in the region below

800nm. Figure 7.11 shows the loss spectra for the Ge-filled ESM PCF structure (shown

500 600 700 800 900

30

60

90

120L

oss (

dB

/cm

)

Wavelength (nm)

ε =i ε /4i

ε =i ε /2i

ε =i εi

Figure 7.11.: Effect of imaginary part of dielectric function on the loss spectra for structureshown in figure 7.3. The curves are plotted for ǫi (blue) ǫi/2 (red) and ǫi/3 (black). The realpart of dielectric function was unchanged in these calculations.

in figure 7.3) in y-polarization. The three different spectra are obtained for three different

fractional values of the imaginary part of the dielectric function of germanium, i.e. ǫi,

ǫi/2 and ǫi/3, while keeping the real part of dielectric function unchanged. It is very

clear from figure 7.11, that for a lower value of ǫi, the coupling is very strong and one

can also achieve the loss peaks even at wavelengths lower than 800nm. Thus, it can be

concluded that for wavelengths < 800nm the loss in the Ge is so high that resonances

cannot form, resulting in an absence of strong loss peaks.

The overlap integral of the total field of glass-core mode on the field inside the Ge-wire

is calculated using finite element simulations. The overlap integral for a specific field

component can be defined as

overlap integral =field inside the Ge-wirefield of the entire mode

(7.1)

For a fixed polarization state, i.e. y-polarization, the value of the overlap integral for the

transverse components of the magnetic field (Hy and Hy) and electric field (Ex and Ey)

144


at some specific wavelengths is plotted in figure 7.12. These wavelengths correspond to

the position of peaks and valleys of the loss spectrum shown in figure 7.6. On comparing

600 800 1000 1200 1400

0.00

0.03

0.06

0.09

Overlap inte

gra

l (m

agnetic fie

ld)

Wavelength(nm)

Hx

Hy

(a)

600 800 1000 1200 1400

0.000

0.001

0.002

0.003

Overlap inte

gra

l (e

lectr

ic fie

ld)

Wavelength(nm)

Ey

Ex

(b)

Figure 7.12.: Ovelap integral of transverse components in y-polarization at some specific wave-lengths. (a) Magnetic field components Hx and Hy (b) Electric field components Ex and Ey.

the position of wavelengths from figure 7.12 and the position of loss peaks from figure 7.6,

it can be observed that the value of field overlap integral is higher for those wavelengths

which observed higher losses. This supports the argument that for these resonance wave-

lengths, the overlap between the Ge-wire mode and the glass-core mode increases due to

phase matching, leading to peaks in the loss spectra of the glass-core mode.

145

Chapter 8Results: metal filled structures

8.1. Gold-filled modified step index fiber

Modified step index fibers (MSIFs) are very useful structures which can be used to in-

vestigate light-metal interactions. Gold-filled MSIFs used in this section were fabricated

by using the direct drawing method, explained in section 6.3.3.

8.1.1. Electrical conductivity measurements

The electrical conductivity measurements were carried out on samples which are fabri-

cated using the technique explained in section 6.4.1. These measurements shows that gold

wires of diameter ≈ 3.8µm in the MSIF are continuous over lengths of several centime-

ters. Individual continuous pieces longer than 5cm could easily be found, which provides

an aspect ratio of ≈ 104. Careful inspection shows that this length of continuous wires

can also be found for wire diameters as small as ≈ 1µm. For smaller diameters ≤ 300nm,

the gold wires show cracks of a few micrometers width, with typical continuous wire-

lengths of several tens of micrometers. For the optical experiments, these cracks proved

to have no significant effect on either the loss characteristics or the spectral locations of

the Surface Plasmon Polariton (SPP) resonances.

147

8. Results: metal filled structures

8.1.2. Optical transmission measurements

Figure 8.1 shows the SEM image of one of the fabricated structure, which is used for

transmission measurements. The filled structure (figure 8.1) consists of a circular gold

wire of diameter ≈ 3.8µm, center to center spacing of ≈ 3.9µm and GeO2-doped elliptical

core (with major and minor axes as 2.9µm and 2.5µm).

2µm

corewire

y

x(b

)

Figure 8.1.: SEM image of gold-filled MSIF polished using focused ion-beam milling. The wirediameter is ≈ 3.8µm. The coordinate system defines the direction of the electric field in twoprincipal states of polarization.

The optical transmission spectrum of the Au-filled structure is obtained using the mea-

surement setup shown in figure 6.19 and explained in section 6.5.1. The polarization

sensitive measurements have been performed by placing both a polarizer and a half-wave

plate before the uncoupling objective as shown in figure 6.19.1

The experimental loss is then calculated for each wavelength, which is shown in figure

8.2. Measurements on an unfilled sample showed that the fiber has a long-wavelength

bend edge at ≈ 1400 nm and a higher-order mode cutoff at ≈ 850 nm. The measured

loss spectrum of the fundamental glass-core mode of the fiber is shown in figure 8.2(a).

At the center of the very strong loss peak at ≈ 1020nm, the cut-back technique results in

noisy curves, owing to the limited dynamic range of the optical spectrum analyzer. No

significant polarization dependence is observed in these experiments.

1The pair of polarizer and half-wave plate is not used in the outcoupled beam for these measurements.

148

8.1. Gold-filled modified step index fiber

900 975 1050 11250

20

40

60

80

Loss (

dB

/cm

)

Wavelength (nm)

X polY pol

900 975 1050 11250

20

40

60

80

Wavelength (nm)

X pol

Y pol902nm

902nm

1011nm

1011nm

(a) (b)

Figure 8.2.: (a) Measured attenuation spectra of light guided in the glass core of the gold-filled MSIF (red, y-polarization; black, x-polarization). (b) Corresponding finite-element sim-ulation of the loss of the MSIF; parameters are given in the text (red: y-polarization; black:x-polarization). The spectral positions of the loss peaks are also mentioned for both the cases.

The finite element simulations2 are then carried out to calculate the loss of the structure.

The dimensions of the core and the hole used to carry out the simulations are the same

as that of the fabricated one (mentioned earlier in this section). The dielectric functions

of the 16 mol.% GeO2-doped silica core and gold were taken from the published data [40]

[80]. In simulations, a mesh size of less than 2nm is used inside the metal adjacent to the

interface so as to accurately resolve the evanescent fields. The simulated loss spectra for

x- and y- polarizations is shown in figure 8.2(b). For both experiments and simulations,

distinct loss peaks appears in the loss spectra of x- and y- polarization at ≈ 902nm and

≈ 1011nm. The spectral positions of the loss peaks matches quite well in experiments

and simulations. It should also be noted that no significant polarization dependence is

observed in experiments.

For x-polarization, the magnitude of the loss in simulations is much higher than in the

experiment, which can be attributed to the excitation of a higher-order mode in experi-

ments. The FEM simulations confirms that this higher order mode in the x-polarization

is the lowest loss glass-core mode, due to a node in the center of the field pattern.3

Excitation of this mode in experiments contributes to the overall transmission and mis-

leadingly reduces the loss for the x- polarization. In y-polarization, however, the mode

(fundamental mode) with a HE11 field distribution in the glass core has the lowest loss

and thus is the only relevant mode in the measurements. As a result, the magnitude of

2By using a commercial software JCMWave.3Axis of the node is the one which connects the center of the Au-filled hole and core of the structure.

149


the simulated loss matches the experiment quite well.

The peak emerging in the simulations at wavelengths beyond 1150 nm (y-polarization)

was not observed experimentally, because the transmitted intensity at those wavelengths

was below the detection limit of the spectrum analyzer. The simulated peak at ≈ 800 nm

was also not observed experimentally, most likely due to being masked by the presence of

higher-order modes. For the 902 nm peak in y-polarization, the simulated and measured

FWHM widths are both ≈ 20 nm, which indicates that the wire diameter is very uniform

(otherwise, the experimental peak would be inhomogeneously broadened). The high

magnitude of the loss for the peak at ≈ 1011nm makes it difficult to experimentally

identify the exact peak position, but for the peak at ≈ 902nm, it is quite clear.

To understand the physics behind the origin of loss peaks, the dispersion of different

guided plasmon modes on a single gold wire embedded in silica is calculated analytically.

The dispersion of the glass-core mode and dispersion of third, fourth and fifth order

guided surface plasmon polariton (SPP) modes are plotted in figure 8.3(a). The crossing

points between the dispersion curves of single guided SPP modes and the curve of the

MSIF core coincide quite well with the spectral position of loss peaks in the loss spectra

shown in figure 8.2, which confirms the excitation of guided SPP modes on the Au-wire

in MSIF. When the real part of refractive index for the single guided SPP mode coincides

with the index of the glass-core mode, energy resonantly transfers from the glass-core

mode into the Au-wire, resulting in strong leakage of power and loss of guidance of the

glass-core.

In order to understand the nature of these SPPs, it can be assumed that planar SPPs

simply spiral around the surface of the wires (as discussed in section 2.2.8). This is valid

when the wires are much thicker than the skin depth in the metal.4 The loss peaks at

902nm and 1011nm can be attributed to the excitation of fourth-order and fifth order

guided modes on Au-wire.

Figure 8.3(b and c) shows the distribution of the axial component of the Poynting vector

of the guided wires at the crossing. These field distributions correspond to the fields on

the Au-wire in FEM simulations. Using the coupled mode theory the coupling lengths

for the fourth-order and fifth-order mode are found to be 1.65mm and 3.8mm.

4If the mode order is high enough, i.e., the angle of spiral is steep enough, then mode index becomesless than the index of the silica host and the SPP mode becomes leaky.

150

8.2. Gold-filled polarization maintaining (PM) PCF

m = 4

m = 3

Real part

of n

eff

m = 5

core

908nm

1010nm

(a)

900 950 1000 1050 11001.460

1.462

1.464

1.466

1.468

1.470

Wavelength (nm)

-0.1000

0.03750

0.1750

0.3125

0.4500

0.5875

0.7250

0.8625

1.000

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

X- coordinate ( m)µ

Y-

co

ord

ina

te(

m)

µ

m = 4

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3m = 5

Y-

co

ord

ina

te(

m)

µ

X- coordinate ( m)µ

(b)

c( )

scale bar

Figure 8.3.: Real part of the effective refractive indices of guided SPP modes on isolated goldwires embedded in fused silica. The numbers refer to the mode order. The magenta curve isthe dispersion of the fundamental glass-core mode. (b) and (c) shows the axial Poynting vectordistributions of the guided SPP modes at the crossing points marked as blue dots (908nm and1010nm).


One of the important PCF structures used to investigate and understand the interaction

of SPP resonances of the gold wire with the glass-core mode is a polarization maintaining

(PM) PCF structure. Schmidt et al. has recently reported the fabrication and optical

characterization Au-filled ESM PCF structures [7]. These filled structures are used to

excite the plasmon resonances at specific optical frequencies.

A PM PCF is a solid-core PCF in which the polarization state of the guided light in the

glass-core is maintained, as explained in section fiber 2.3.1.1. If designed properly, the

glass-core of the PM PCF will only support the fundamental mode of any polarization for

all wavelengths in visible and near IR. To investigate the SPP resonances, only the single

hole adjacent to the core is filled with gold using the selective hole treatment technique

(section 6.2.3) and filling techniques (section 6.3). Figure 8.4 shows the SEM image of

an Au-filled PM PCF, in which one of the small elliptical holes adjacent to the glass-core

is filled with gold (Major and minor axis dimension 1.23µm and 0.85µm). The fiber

consists of a rectangular glass-core with 3.57µm and 1.1µm as length and width. The

optical transmission spectra of the glass-core mode of the filled structure is measured

using the techniques explained in section 6.5.1. In this case, both a polarizer and a half-

151


x

y

2 mμ

Figure 8.4.: SEM image of a small hole PM PCF, in which a single elliptical small hole (0.85µmand 1.23µm as minor and major axis, respectively) adjacent to the core is filled with gold. Therectangular core of PM PCF consist has dimensions of 3.57µm and 1.1µm as long and shortside of rectangle.

wave plate are placed before the incoupling objective lens, which allows light to couple

only single polarization state of the glass-core in PM PCF.

Figure 8.5(a) shows the transmission spectra of the glass-core mode for the y-polarization

of the PM PCF for two different lengths of the fiber, i.e. 6mm (blue curve) and 24.5mm

(orange curve). Distinct transmission dips appear were observed at three specific wave-

lengths ( 620, 690, and 890nm). In order to understand the origin of these dips, the

dispersion curves (Re(neff ) versus wavelength) of different guided SPP modes (of a sin-

gle Au-wire (diameter 0.85µm) embedded in silica), PM PCF glass-core and silica are

plotted in figure 8.5. The dispersion curves for guided SPP modes (2nd, 3rd, 4th and

5th oder) were calculated analytically. The dispersion of unfilled PM PCF was modeled

using finite element technique5. For the purpose of simplicity all the holes were assumed

to be perfectly circular in FEM simulations.

In figure 8.5(b), the glass-core mode (black curve) extends into the air holes, which lowers

its effective index below that of silica (red curve). It is interesting to observe that the

real part of the indices of the SPP modes can extend below the silica line if the imaginary

part of the dielectric function of gold is large, which also has the effect of shifting the

modal cutoff points.

5Using commercial software JCMWave.

152


-60

-50

-40

-30

-20

-10

0

Tra

nsm

issio

n (

dB

)

4

3

2

A

B

500 600 700 800 900 10001.44

1.45

1.46

Wavelength (nm)

silica line

unfilled PCF

2

345Re

al p

art

of

neff

6mm

24.5mm

Y-pol (a)

(b)

Figure 8.5.: (a) Measured transmission spectrum in y-polarization of a 6mm (blue curve) and24.5mm (orange curve) of length of the structure shown in figure 8.4. The mode number markedon each transmission dip corresponds to the modes of the gold wire to which the glass-core modeis phase matched. (b) The dispersion curves of pure silica (red curve), unfilled PCF (black curve)and different guided SPP modes of a single Au-wire embedded in silica. The different Au-wiremode orders are marked with their corresponding mode numbers, i.e. 2, 3, 4, 5.

153


The dips in transmission shown in figure 8.5(a) correspond to anti-crossings between

the core mode and successive SPP modes of the 3rd and 4th order on the wire, the

experimental dips being shifted slightly to shorter wavelengths. The dip corresponding

to the coupling of 5th order SPP mode was too weak to be seen in the measurement. The

shift of dips to shorter wavelengths can be explained by the presence of narrow air gaps

between the wire and the glass, caused by more rapid thermal shrinkage of the metal

[7]. The non-circularity in the wire cross-section in experiments may also play a role.

However, the 2nd order mode in experiment dip does not match well to the modeling and

there being no apparent anti-crossing at 890 nm. It can be attributed to the fact that a

greater SPP modal area at longer wavelengths causes the SPP mode to increasingly feel

the effects of the surrounding air holes (which is not included in the analytical model

for a wire embedded in silica), which lowers its effective index and shifts the predicted

anti-crossing point to a shorter wavelength experiments.

The orange curve shown in the transmission spectrum for y-polarization (figure 8.5(a)) is

for a sample length of 24.5mm. The dip for 2nd order mode is 45dB down in this sample,

which is much deeper than previously reported devices of similar length, e.g., long-period

PCF gratings. These results suggest that gold-filled PCFs could also be potentially used

as an effective in-fiber filter.

The argument that coupling with the SPP modes on the Au-wire causes the dips in the

transmission spectra (figure8.5(a)) is further supported by imaging the near-field images

of fiber output end face. These images were taken by first using a 60X objective lens to

outcouple the light from the filled sample and then imaging it over a a charge coupled

device camera6. Notch filters with a bandwidth of 10 nm (much narrower than the SP

resonances) were used to image the near field at several wavelengths.

The results for a 6 mm length of the filled PM PCF at wavelengths 546 nm and 907 nm

are shown in figure 8.6. For a clear understanding of these two wavelengths with respect

to positions of the transmission dips, these wavelengths are marked as point A and B in

figure 8.5(a).

In order to understand the coupling of light from the glass-core mode to the SP resonances

of the wire, optical image of the output end face of the filled structure is overlaid on top

of the SEM of the filled fiber. At 546 nm, the mode is confined to the core, with no

light being visible on the wire due to the weak coupling strength and the wavelength

6uEye camera for visible wavelengths.

154


A: 546 nm B: 907 nm

Figure 8.6.: Experimentally observed optical near field images of the fiber end face at twodifferent wavelengths (A) 546nm and (B) 546nm, superimposed onto the SEM image of thefilled PM PCF. The position of the wavelengths are also marked in transmission spectra shownin figure 8.5.

offset between the filter and the SPP resonance. For a filter wavelength of 907 nm,

however, almost all the light is concentrated on the wire. Since, 907 nm corresponds to

the position of the 2nd order SPP resonance on the wire, this near-field image confirms

the excitation of a SPP mode, which travels along the wire, scattering strongly when it

reaches its protruding end. Interestingly, there is no light in the core in this case. The

coupled-mode theory reveals that the distribution of power between glass-core mode and

SP mode depends on the precise detuning from phase synchronism which varies with

wavelength, the loss level in the wire and the interaction length. Situations can be found

in which the light is entirely on the wire (at 907nm in figure 8.6), is shared between core

and wire, or is entirely in the core.

The polarization dependent characteristics is then investigated by measuring the trans-

mission spectrum for x- and y- polarizations. Figure 8.7(a) shows the measured trans-

mission spectra and figure 8.7(b) shows corresponding simulations. In both cases the

transmission in x-polarization is lower (up to 15dB in experiments) in comparison to the

y-polarization. This higher loss for x-polarization can be attributed to the fact that the

power absorbed at a metal surface is proportional to the component of magnetic field

which is tangent to the surface (similar to germanium). The small ripples in the trans-

mission spectra for x-polarization are due to the presence of the unavoidable excitation

of cladding modes, which is noticeable because the signal from the glass-core mode is

weak. The spectral positions of the dips in experiments shifted to lower wavelengths

155


500 600 700 800 900 1000

0.01

0.1

1

Tra

nsp

are

ncy (

cm

/dB

)

Wavelength (nm)

Y-pol

X-pol727nm

975nm

660nm

500 600 700 800 900 1000-50

-40

-30

-20

-10

0

X-pol

Tra

nsm

issio

n (

dB

)

Wavelength (nm)

Y-pol

727nm 975nm660nm

Figure 8.7.: (a) Measured transmission spectra (in dB) for the two polarization stateS for thestructure shown in figure 8.4. (b) The FEM simulation of the structure. For convenience thetransparency (cm/dB) of the glass-core mode is plotted for simulations. The dashed line infigure (a) corresponds to the positions of the dip in simulations.

in comparison to the FEM simulations. This shift to lower wavelengths can again be

attributed to a small air gap [7].

Another important feature for these structures is the device length. The cut-back mea-

surements were made on these type of structures to estimate the loss [6]. The observed

value of the loss was reported to be ≈ 2.5dB/mm for wire diameter of 600nm, which is

lower than the previously reported value in Au-filled ESM structures [7]. The transmis-

sion characteristic of these Au-filled PM PCFs devices can be used to form an in-fiber

wavelength-dependent notch filter.

156

Chapter 9Device applications: germanium-filled

structures

Using the optical and electrical properties of germanium, Ge-filled PCF/MSIFs can be

used to fabricate in-fiber devices and sensors. Apart from that, the high nonlinearity of

germanium can be used to generate a supercontinuum (SC)light source for mid and far

infrared region.

9.1. In-fiber temperature sensor

The temperature dependence of optical properties of germanium is explained in detail

in sections 5.2 and 3.4. The dielectric function and refractive index of germanium is

a function of temperature, indicating that the resonance wavelength of glass-core and

Ge-wire mode is also a function of temperature. This temperature dependence of the

resonances is used to realize a in-fiber temperature sensor.

The temperature dependence of the loss peaks in the optical transmission measurements

(shown in figure 7.6) for the 0.8 mm long device structure is depicted in figure 7.3 . The

experimental setup for this measurement is similar to the one which is used to measure

the optical transmission with an additional arrangement for temperature variation of the

sample. A Peltier element is attached to the fiber holder in order to set temperatures in

the range 295 to 343K and then the transmission spectrum for a particular temperature

is measured using the OSA. The measurements are focused on TE05 (1180nm), EH15

157

9. Device applications: germanium-filled structures

(1300nm) and EH24 (1390nm) peaks in y-polarization, since they have lower loss and

consequently very good spectral visibility (figure 7.6). Figure 9.1 shows the measured

05

15

24

Wavele

ngth

shift (%

)

Temperature difference (K)

0.60

0.45

0.30

0.15

0.0030 35 40 45 500 5 10 15 20 25

0.13 nm/KTE

EH 0.18 nm/K

0.15 nm KEH

Figure 9.1.: Measured relative spectral shift of the position of three of the resonance peaks(TE05 (1180nm), EH15 (1300nm) and EH24 (1390nm) peaks in y-polarization) is plotted versustemperature. The slopes of the fitted linear curves are shown in units of nm/K.

percentage shift 100(λT − λ295)/λ295 of each peak relative to its spectral position at

ambient temperature. As the temperature rises, the loss peaks shift to longer wavelength.

The rate of shift is 0.13 nm/K (TE05), 0.15 nm/K (EH15) and 0.18 nm/K (EH24). The

physical origin of this shift lies in the fact that with the increasing temperature the

indirect electronic band gap of Ge decreases (figure 3.9 in section 3.3.2). This shifts the

absorption band to a longer wavelength (figure 3.13 in section 3.4), which then leads to

an increase in the real part of the dielectric function in the IR region due to Kramer-

Kroenig’s relations (section B) [28].

The increase in temperature can cause change in the refractive index of germanium but it

can also cause the thermal expansion of the Ge-wire. In order to estimate and understand

the effect of these two parameters, finite-element calculations were performed for two

cases; (i) the increase of the wire diameter due to the thermal expansion of germanium,

and (ii) the change in the dielectric functions for both Ge and silica with temperature.

The shift in the position of loss peaks only due to the thermal expansion of wire (in the

temperature range of experiment) was found to be very small in comparison of observed

shift in experiments. When the change of the dielectric functions for both Ge and silica

with temperature is included, the shift in the positions of the loss peaks is larger. This

158

9.2. In-fiber photodetector

proves that the major contribution in this shift is due to the change in the dielectric

function of germanium with temperature. The shift in the loss peaks were then used to

evaluate the slope dλ/dT for the resonances. The simulations yielded slopes of 0.36nm/K,

0.30nm/K and 0.37nm/K for the TE05, EH15 and EH24 modes, respectively. These values

are higher than the experimental ones (see the plot in figure 9.1). This can be attributed

to the temperature dependent changes in the stress in the composite Ge:silica structure.

This is quite likely to be the case, since Ge expands when it solidifies from the liquid

state[23]. Also, there would be a slight discrepancy in the measured temperature of the

Peltier element (which is used in simulations) and the temperature within the wire.

The measured sensitivity in the Ge-filled in-fiber temperature sensor can be compared

with that of obtained in other fiber-based devices. Bragg grating sensors provide ≈0.01

nm/K [24], and long-period gratings are somewhat more sensitive, yielding ≈ 0.1 nm/K

[25, 26]. The use of hybrid structures can considerably enhance the sensitivity in each

case, up to 19 nm/K (over a < 2 K range) in the case of long-period gratings in standard

fiber [27] and ≈ 10 nm/K in hollow-core PCF filled with fluids or liquid crystals [28, 29],

although the temperature range over which such hybrid devices operate is often quite

restricted. The length of the device is also an important consideration and the 0.8 mm

long Ge-filled PCF device leads the field. For comparable sensitivities, it is between 10

and 100 times shorter than the Bragg grating and long-period grating devices reported

in the literature, which are typically between 1 and 10 cm long.


One of the important opto-electronic properties of germanium is the intrinsic photocon-

ductivity, as explained in section 3.4.3. The optical measurements discussed in section

7.2 shows that these Ge-filled PCF/MSIF structures are excellent hosts for investigating

the interaction of light with germanium. These two observations can be combined to

fabricate an in-fiber photodetector.

9.2.1. Side pumping mode

In one experiment, the charge carriers, i.e. electron and holes, were generated inside the

germanium wire by focusing light through the side of the fiber. The experimental setup

159


used to observed this effect is discussed in section 6.5.2.1. First, the experiment was

performed on a single capillary with Ge-wire diameter of ≈ 10µm. A very small (difficult

to measure accurately) photocurrent was observed using the incident light of 532nm. To

enhance this effect, a solid-core PCF structure with hole diameter ≈ 1µm and pitch ≈7µm is used experiments. This structure is termed as large pitch PCF (LP PCF). The

variation of photocurrent with respect to incident power is measured for different applied

voltages.

0.00 0.05 0.10 0.15 0.200.005

0.010

0.015

0.020

0.025

0.030 5 V7 V

10 V

Ph

oto

cu

rre

nt

(A

)μ

Incident power (W)

λ= 532nm

Figure 9.2.: Variation of photocurrent with incident power for a Ge-filled large pitch PCF withhole diameter ≈ 1µm and pitch ≈ 7µm. The experiments were performed for three differentcircuit voltages.

Figure 9.2 shows the variation of generated photocurrent with the incident power for three

different applied voltages. For all three applied voltages, the generated photocurrent first

increases linearly with incident power and then starts to saturate for higher power levels.

Initial increase in the photocurrent with incident power can be attributed to the increase

in the photo-generated charge carriers at higher power levels.1 As the illuminated area of

Ge-wires is constant, further increase in the number of incident photons (which is related

to power) does not generate more charge carriers, as there are no electrons remaining to

get excited with this energy, which ultimately leads to the saturation of photocurrent.

The incident power at which the saturation of photocurrent occurs decreases with increase

in the applied circuit voltage. For higher applied voltages, the charge carries which are

1Increase in the power increases the number of incident photons in the illuminated area, which leadsto an increase of the number of photo-generated carriers.

160


generated low powers can acquire sufficient acceleration to contribute in the photocurrent,

as a result the saturation occurs at low incident powers.

The numerical estimation of the photocurrent in these structures is difficult in this ex-

periment due to the presence of more than one wire. Furthermore, a large amount of

incident light is reflected and scattered from the fiber surface and the Ge-wire and silica

surface boundary. Focusing of the light on the Ge-wires is also difficult due to the curved

outer surface of the fiber.

9.2.2. Transmission mode

The absorption of light in the Ge-wire while transmitting the light through the core in

Ge-filled MSIF sample can be used to fabricate an in-fiber photodetector in transmission

mode. This method of measuring the photoconductivity provides a better efficiency to

generate the charge carriers, i.e. electron and holes. The experimental setup to measure

the photoconductivty in transmission mode is discussed in detail in section 6.5.2.2. Figure

2 mµ

core

Ge-wireY pol

X pol

543nm

500 600 700 800 900-75

-70

-65

-60

-55

-50

Tra

nsm

issio

n (

dB

m)

Wavelength (nm)

(b)(a)

Figure 9.3.: (a) SEM image of the MISF strcture (core diameter ≈ 1µm and Ge-wire diameter≈ 1.3µm) used to measure photoconductivty. (b) Transmission spectrum (in dBm) in x- and y-polarizations for the structure shown in (a).

9.3(a) shows the SEM image of the MSIF. The MSIF used in this measurement has a

GeO2-doped silica core with a diameter of ≈ 1µm and a Ge-wire with diameter of the ≈1.3µm. The OD of the fiber is ≈ 220µm and the length of the filled sample is ≈ 1.5cm.

An ITO layer of ≈ 30nm is sputtered on both the ends of the fiber, while an 8mm space

in the middle of the sample is kept un-sputtered.

161


As a first step, the transmission spectrum of the Ge-filled MSIF is measured both in x-

and y- polarization, as shown in figure 9.3(b). The transmission in x-polarization is lower

than in the y-polarization, because tangential magnetic field lies parallel to the surface

for x-polarization, which leads to higher absorption of light in x-polarization (discussed

in detail in section 7.2.2). In the visible wavelength range from 500nm to 600nm, the

spectrum is free from any spectral features. For wavelengths greater than 600nm there

are various ripples in the spectrum, which is due to the presence of light from cladding

modes. For these wavelengths, the light coming from the glass-core mode is extremely

weak and could not be measured efficiently. The fiber structure is designed in such a way

that it has single mode guidance a from 400nm to 1000nm.

Below below 575nm, germanium has a quantum efficiency (to generate charge carri-

ers) which is proportional to the total absorbed energy [81]. Based on this fact, light

at 543nm2 is chosen to generate the charge carries. Another reason of choosing this

wavelength is related to the transmission characteristics of the fiber. The transmission

spectrum of the fiber (figure 9.3) shows that at 543nm the power of the incoupled and

outcoupled light from the glass-core mode can be measured efficiently and the difference

between the transmission of x- and y- polarizations is also significant. For higher wave-

lengths of the He-Ne laser (633, 612, 604nm), the transmission for x-polarization is quite

low and the measured light from the core could have contributions from the cladding

light. The difference of input and output light must be accurately measured because it

used to estimate the absorbed power in the Ge-wire.

In experiments, the value of generated photocurrent Iph is obtained by taking the diffe-

rence of current measured with and without the presence of light in the core

Iph = Ilight − Inolight (9.1)

The experimentally measured value of the generated photocurrent as a function of ab-

sorbed power is shown in figure 9.4. In these measurements the applied circuit voltage

is 20V. It is clear from this figure that, with the increase in the absorbed power, the

value of the generated photocurrent increases linearly. Two important conclusions can

be made from figure 9.4

(i) the value of the photocurrent increases linearly with power. As light is guided through

the glass-core, photons get absorbed in the Ge-wire due to the field overlap between the

2It was also the shortest available wavelength respect to 575nm, among all the available wavelengthfrom continuous He-Ne lase system (section 6.5.2.2).

162


0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

0.25

0.30

Absorbed power ( W)µ

X-pol

Photo

curr

ent (n

A)

8.45 pA/ Wµ

Y-pol

7.39 pA/ Wµ

λ= 544nm

Figure 9.4.: Change in photocurrent (in nano-amperes (nA)) as a function of absorbed power(in µW) in the Ge-wire for x- and y- polarizations of the incident light. The straight lines arethe linear fit to the measured data points.

glass-core and the Ge-wire, which generate new charge carriers, i.e. electrons and holes.

Under the application of the electric field (due to circuit voltage of 20V), these charge

carries generate an additional current called the photocurrent. With the linear increase

in absorbed power, the number of photons absorbed in the Ge-wire increases linearly3,

which causes a linear increase in the photocurrent.

(ii) the magnitude of generated photocurrent is larger for x-polarization in comparison

to y-polarization. The measured slope of the data points for x- and y- polarizations is

8.45pA/µW and 7.39pA/µW , respectively. The higher slope in x-polarization indicates

that more charge carriers are generated in x-polarization. This can be attributed to the

higher absorption of the light in the in x-polarization in comparison to the y-polarization

as shown in transmission spectra (figure 9.3(a)).

Equation 3.55 (section 3.4.3) is used to numerically estimate the value of photocurrent

per unit power in MSIF. In this case, the sample thickness is equal to the diameter of

the wire (≈ 1.3µm) and αabs is ≈ 0.5 × 10−4 m−1 at 543nm . The volume of the wire

is π(d/2)2l and area of the electrode is the area of the end face of the wire π(d/2)2. If

mobilities of electron and hole are µn = 3800cm2/Vs and µp = 1800cm2/Vs, carrier life

time4 is ≈ 1µs and length of the sample is 1.5cm, then the empirical relation between

3number of photons = Pabs/hν, for a constant time.4For pure germanium at 300K.

163


the photocurrent Iph and absorbed power Pabs can be expressed as

Iph = 0.0281× Pabs (9.2)

here, Iph is in Ampere and Pabs is in Watt. Using the expression in equation 9.2, the value

of the generated photocurrent can be estimated for a particular value of absorbed power5.

The estimated value of the slope was found to be 28.1 nA/µW , which is approximately

three orders of magnitude higher than the measured values of the slopes (figure 9.4).

In this experiment, the incident light on the Ge-wire is the light which is coupled onto

the wire due to a filed overlap between core mode and Ge-wire. In an ideal case (no

coupling and scattering loss), this light can be considered as the difference between the

incoupled light to the outcoupled light of the MSIF-core. In practical situations there is

coupling loss which can allows only 10 % of incident light to be couples into the core 6.

If this coupling loss is included in calculations, the efficiency of photocurrent generation,

i.e. the value of numerical estimated slope is 2.81 nA/µW .

As explained in section 3.4.3, the value of the photocurrent can be significantly reduced

due to the presence of surface states as well as the barrier height due to difference

work function of ITO and Ge. The work function of sputtered ITO can be taken as

≈ 4.26eV [82] and for germanium ≈ 4.506eV [83]. In this experiment, no annealing

was performed to avoid any effect of diffusion of ITO in silica, which can change the

transmission characteristics of the fiber. Due to these reasons the ITO-Ge interface does

not work as an ohmic contact and would reduce the generated photocurrent.

The germanium inside the MISF is also not single crystalline (section 7.1.2), the charge

carries can get trapped in the grain boundaries and defect centers which also obstructs the

flow of charge carries. The table summarizes the affect of these factors on the efficiency

of the photodetector, Despite all these limitations which reduces the effective value of

generated photocurrent, the measured value of the photocurrent allows the Ge-filled

MSIF to be considered as a good candidate for an in-fiber photodetector.

The effective change in the resistivity of the Ge-filled MSIF can also be obtained by

measuring the difference in the value of resistivity with and without the presence of the

photocurrent. The resistance R (which is directly obtained from the HP resistivity meter)

5This expression is derived under the assumption of perfect ohmic contact of germanium with externalcircuit.

6Due the small core of the MSIF the coupling loss is higher.

164


Efficiency of detector

Perfect photodetector 28.1 nA/µW

Including coupling loss 2.81 nA/µW

Experimental 8.45pA/µW and 7.39pA/µW

Table 9.1.: Efficiency of the photoconductor.

is related to the resistivity ρ as

ρ = Rπr2

l(9.3)

here, r and l is the radius and length of the Ge-wire respectively. In this case, radius is

0.65µm and length is 1.5cm. Figure 9.5 shows the change in the resistivity ∆ρ (in Ω-cm)

as a function of absorbed power for both x- and y-polarizations.

Re

sis

tivity c

ha

ng

eΔ

?(Ω

-cm

)

Y-pol

Absorbed power (μW)

X-pol

0.4859 Ω μ-cm/ W

0.5214 -cmΩ μ/ W

λ= 544nm

0 5 10 15 20 25 30-2.0

-1.6

-1.2

-0.8

-0.4

0.0

Figure 9.5.: Change in resistivity ∆ρ (in Ω-cm) as a function of absorbed power in the Ge-wire for x- and y- polarization of incident light. The straight lines shown the linear fit to themeasured data points.

As expected, the resistivity of germanium decreases with increase in absorbed power

(described by the negative sign of the axis in figure 9.5). The magnitude of the change in

the resistivity can be expressed by the slope of the data points, which is 0.5214Ω−cm/µWand 0.4859Ω − cm/µW for x- and y-polarizations, respectively. The higher value of

the slope for x-polarization corresponds to the higher value of the photocurrent in x-

polarization.

165


Above discussion clearly shows that Ge-filled MSIF worked as an in-fiber photodetector,

which can find possible applications in in-fiber devices.

9.3. High nonlinearity application of a germanium

waveguide: theoretical analysis

One of the important optical property of the germanium is its optical transparency in

the infrared region (2µm to 16µm), which can be used to make a step index optical

waveguide (similar to a conventional optical fiber for visible and near IR) for broadband

IR region. Another additional feature of the germanium is its nonlinear refractive index

n2 of the order of 10−15 m2/W [70], which is at least 5 order of magnitude higher than

silica (≈ 10−20 m2/W). This high nonlinearity and IR transparency of germanium can

be combined to fabricate a waveguide which can be used to generate a broad band light

source for IR region. In a Ge-waveguide, due to its high nonlinearity a very small length

(≈ 2-5mm) and relatively low peak power of the pulse may be sufficient to generate SC

source in IR.

The supercontinuum generation of light using optical fibers has been a interesting area

of research since 1961. The SC generation in visible and IR is discussed in detail by

Dudley and Taylor [84]. Silica based PCFs were used to generate SC in visible and near

IR region [75] [85], while in case of mid and far IR region optical fibers based on other

glasses were used to generate the SC light sources e.g. chalcogenide, tellurite [86], [87].

The main motive to fabricate a Ge-waveguide is that the high third order nonlinearity

of germanium can be used to generate new frequencies.

The fabrication techniques used in this work provide a unique opportunity to fabricate

Ge-waveguide with small diameters (≈ 500nm has already been achieved for germa-

nium). The most important parameter of a waveguide for generating SC is the zero

dispersion wavelength (ZDW) (section 2.2.3.1). A high energy pulse propagating near

ZDW in Ge-waveguide can use nonlinear phenomenas such as self phase modulation,

soliton generation-dispersive waves and four wave mixing to generate new frequencies.

For germanium it may require a very short length of wavelength due to its high nonlin-

earity. Figure 9.6(a) shows the variation of dispersion coefficient D ((section 2.2.3.1))

with respect to wavelength for step index waveguide with a Ge-core and silica cladding.

The calculated plots were obtained for the fundamental mode of the waveguide. This

166

9.3. High nonlinearity application of a germanium waveguide: theoretical analysis

1000 1200 1400 1600 1800 2000

-8000

-4000

0

4000

8000

320nm

320nm

Dis

pe

rsio

n c

oe

ffi. (

ps/(

nm

-km

))

Wavelength (nm)

340nm

290nm

(a)

1500 1600 1700 180010

-7

10-4

10-1

102

Lo

ss (

dB

/mm

)

Wavelength (nm)

340nm

320nm

320nm

290nm

(b)

Figure 9.6.: Calculated dispersion coefficient and loss for a step waveguide with puregermanium-core and silica-cladding (a) Dispersion coefficient (in ps/(nm-km)) (b) Loss in(dB/mm).

waveguide can guide light due to index guiding because the index of refraction of ger-

manium (≈ 4) is higher than silica (≈ 1.4). These plots helps to estimate the diameter

which gives a zero dispersion wavelength close to commercially available pulsed laser sys-

tems in near IR. The calculated loss spectra of such an waveguide is also shown in figure

9.6(b), which does not change substantially with increase in core diameter. A Ge-core

with diameter of ≈ 290nm is necessary to obtain a single mode guidance with a ZDW

near 1550nm. This waveguide with ZDW near 1550, can be used to pump it with the

commercially available 1550 pulsed laser system and generate new frequencies using the

high nonlinearity of germanium. It is clear from above discussion that it is possible to

predict the Ge-waveguides, which can be used as a small device to generate SC in mid-

and far- infrared region.

As a first step to realize such structures, transmission measurements were performed on

Ge-filled capillaries of different wire diameters to achieve light transmission through the

Ge-core. Although the Ge-wire used in the transmission experiments are continuous, the

fabricated samples so far has not shown any guidance behavior in core. The main reason

of this non guiding core could be attributed to the polycrystalline nature of material inside

the capillaries (micro-Raman experiments in section 7.1.2). Also the Ge-wire inside the

capillary experienced a stress which can change the transparency characteristics of the

germanium (section 3.4.1.6). Further investigation to understand the structural behavior

of the germanium is necessary to optimize the filling process to achieve the guiding Ge-

core.

167

Chapter 10Conclusion and Outlook

The main objective of this work was to fabricate semiconductor/metal -filled PCF/MSIF

samples and analyze their optical properties. These novel structures provide a unique op-

portunity to demonstrate some in-fiber devices. The result of this work can be concluded

by formulating answers to the questions introduced in chapter 1.

Is it possible to fill PCFs with semiconductor materials and metals ?

The techniques discussed in chapter 6 allowed us to fabricate high purity and high optical

quality gold as well as germanium-filled PCF/MSIF structures. The pressure cell and

splicing filling techniques were modified for germanium in order to prevent the oxidization

of germanium during the filling process. The splice filling method was further optimized

in order to obtain smaller wire diameters, below 100nm for gold and 500nm for germanium

with length-to-diameter ratios as high as 10,0000. One important technique developed

and optimized during this work is the filling of only selected holes in a PCF. This allows

control over the spectral loss of glass-core mode, as well as introduces birefringence into

the structure. The fabrication technique for direct drawing of Au-filled MSIF structures

was also developed in this work. This technique helps to obtain several meters of the

filled samples in a single drawing process, which are much longer then the useful length

(≈ 1cm) for any transmission experiment.

The micro-Raman measurements have been performed to obtain information about the

structural properties of Ge-filled samples, which shows a symmetric peak at 297.6cm-1.

This indicates that the germanium inside the sample is highly crystalline but not single

crystalline. In Au-filled structures, the conductivity measurements confirmed the appear-

169

10. Conclusion and Outlook

ance of gold columns (of length ≈ 5cm) that are separated by few microns. This effect

is more pronounced for smaller wires (diameter < 500nm). In contrast, the conductivity

measurements for the case of germanium confirmed the absence of gaps in the Ge-wire

because of its interesting property of expansion during the solidification.

In conclusion, it was possible to successfully fill semiconductor materials and metals into

hollow channels of PCF/MSIF. These filling techniques still need further improvement in

order to reduce the number of gaps in the Au-filled structures. The filling technique must

also be improved to achieve a waveguide with a pure germanium core. A diameter of ≈300nm is necessary to achieve single mode guidance in these pure germanium waveguide.

What are the effects of semiconductors and metals wires on the transmission

characteristics of PCFs glass-core mode ?

The optical transmission spectra of the fundamental mode of the glass-core in PCF/MSIF

confirms the coupling of light from the glass-core to the resonances of the Ge/Au wires.

For Ge-filled PCF structures, pronounced dips were observed in the transmission curve

(which corresponds to the peaks in loss curve) of the glass-core mode of the PCF. In

order to control the magnitude of loss in visible and near IR regions, optical transmission

spectra were measured in a sample which has a single wire adjacent to the core of an ESM

PCF. This introduces birefringence in the ESM PCF, as well as causes strong polarization

dependent transmission losses, with extinction ratios as high as 30 dB in the visible

region. In the IR region, anti-crossings between the glass-core mode and Mie-resonances

of the Ge-wire create a series of clear dips in the spectrum transmitted through the fiber.

The measurements agree closely with the results of finite-element simulations in which

the wavelength dependence of the dielectric constants is fully taken into account. On

comparing the mode field pattern obtained from the analytical calculations of a Ge-wire

embedded in silica, with the fields in the wire from the FEM simulation, mode orders

were identified. A simple model based on a multilayer structure is also used in this work

to help interpret the results [5].

For gold-filled PM PCF (or ESM PCF structures as reported by [7]) structures, the

coupling of light from the glass-core to the plasmonic resonance on the wire leads to the

presence of prominent peaks in the loss spectra of the glass-core mode. The spectral

position and origin of the observed peaks in experiments were in good agreement with

the corresponding analytical calculations and FEM simulations. The spectral positions

of the measured loss peaks were found to be shifted slightly towards shorter wavelengths

170

due to the presence of small air gaps between the surface of gold and silica. This is

attributed to the mismatch in the thermal expansion coefficients of silica and gold.

The overall optical loss of the glass-core mode consist of the loss caused by the field

overlap of the glass-core mode with the plasmonic modes on metal wire and the loss

peaks at specific spectral positions. The pronounced loss peaks observed in the spectra

are due to the phase matching between the glass-core mode and SPP modes on gold wire

at these wavelengths. The optical imaging of the end face of a sample with a single Au-

wire adjacent to the core of a PM PCF confirmed the origin of loss peaks to be plasmon

resonances of the wire. A spiraling plasmon model proposed by Schmidt et al. [7] [20]

was used to describe the propagation of these plasmonic modes of the wires.

What are the possible fiber structures (other than PCFs) which can be fab-

ricated to investigate the light-matter interactions?

In the quest of designing new fiber structures that provide low loss guidance in the core

as well as efficient coupling from the glass-core mode to the modes of the Ge/Au wires,

a modified version of conventional step index fiber (MSIF) was designed by introducing

a hole adjacent to the GeO2-doped glass-core. The coupling of light from the glass-core

to the resonances of the wire was also observed in filled MSIF structures. An important

advantage of these structures is that they are mechanically more stable in comparison

to PCF after the filling. One other important improvement of Au-filled MSIF structures

over PCFs is that the direct drawing of these filled structures was found to be relatively

easier in comparison to the drawing of a filled PCF structure. This is due to the presence

of only one hole. Additionally, the difficulty of selective hole filling is automatically

removed in these structures. The simulations and analytical calculations were found

to match quite well with the spectral positions of the measured loss peaks of the Au-

filled MSIF directly drawn from the fiber tower [74]. The precise control over the fiber

dimensions was achieved by controlling the drawing parameters.

What kind of in-fiber devices can be realized using these novel structures?

The temperature dependence of the optical properties of germanium indicates that the

Ge-filled structure can be proposed as an in-fiber thermometer. The ESM PCF with a

Ge-wire adjacent to core has shown a maximum shift of 0.18 nm/K in spectral positions

of one of the loss peaks with temperature.

In this work, experiments were performed for a temperature difference of ≈ 50K but

these structures can potentially be used to operate in the temperature range of ≈ 200K.

171

10. Conclusion and Outlook

Therefore, the performance of Ge-filled ESM PCF temperature sensors is predicted to

be as good as to the other grating based in-fiber temperature sensors. One area in which

Ge-filled sensor showed significant advantage over grating based sensors is the device

length, which is only 0.8mm in Ge-filled ESM PCF, making it 10 to 100 times shorter

than other grating based in-fiber sensors.

The ability of germanium to generate photo induced charge carriers was used to fabricate

an in-fiber photodetector. ITO was used as a electrical contact material to connect the

Ge-wire with an external circuit. The generated photocurrents for the two orthogonal

polarizations of incident light, show efficiencies of ≈ 8.45pA/µW and 7.39pA/µW . In

absence of any loss mechanisms (for ideal case) efficiency was found to be three orders of

magnitude higher then the measured one. Further improvement of efficiency is predicted

when a thicker layer (≈ 400nm) of ITO is deposited on the Ge-wire. The quality of the

contacts can be further improved by annealing of the sample with correct parameters [88].

Other suitable materials such as In, AuGa, AuSb that makes ohmic contacts with Ge,

can be tested in order to increase the efficiency of the device. The detailed investigation

of the structural properties of the germanium followed by optimization in filling tech-

niques would also help to improve the efficiency of Ge-filled PCF/MSIF in-fiber devices.

Nevertheless, these experiments demonstrated the utility of such novel structures.

The fabrication of a Ge-waveguide still poses a challenge for the developed techniques.

Further improvements in fabrication techniques or post filling treatment of the Ge-filled

samples are necessary to achieve light guidance in a pure germanium core. Although

fabrication of Glass-clad single-crystal germanium optical fiber has been reported but no

successful transmission has been achieved so far.[89]. Due to the high third-order nonlin-

earity of germanium, such Ge-core waveguides will provide an opportunity to generate

broadband sources for mid and far IR. The Au-filled PM PCF can be used as a potential

wavelength-dependent notch or as a polarization filter. Fibers with a single metal wire

sticking out of the end face could be used as near-field tips for the subwavelength scale

imaging by exciting a plasmon on this metal wire tip. These fabrication techniques can

also be applied to design a structure which is useful in ion-trap applications. The tech-

niques discussed in this work has already been used to fill soft-glasses in silica capillaries

[90]. Step waveguides of chalcogenide or tellurite glasses have been used in the gener-

ation of supercontinuum in IR region. A PCF structure with a small Au-filled hole in

the center of the solid-core fiber can be used to create a modal filter, which transmits an

azimuthal polarized light from a random input polarization state.

172

Appendix ASplicing parameters

Prepush (µm) 5

Pregap (µm) 5

Hot push (µm) 13

Push velocity (steps/s) 650

Hot push delay (s) 0.1

Power (Watts) 14-18

diameter((µm)) 1-6

Duration(s) 2.1

Table A.1.: Splicing parameters to splice a capillary of inner diameter of 150µm with a MSIFwith 1.0µm hole

173

Appendix BKramer-Kroenig’s relations

The real ǫr and imaginary ǫi parts of the complex permittivity ǫ of the medium are not in-

dependent of each other. They are related through Kramer’s Kronig relations. Kramer’s

Kronig relations are the mathematical functions connecting the real and imaginary parts

of any complex function which is analytic for the upper half plane i.e positive imaginary

part. In general they can be applied to any response function of time varying linear

passive system and relate the real and imaginary part of its frequency response function.

One important condition which is required for the Kramer’s Kronig relation is that the

response of any impulse must decay in time. Under the assumption the ǫr and ǫi can be

expressed as

ǫr(ω) =1

πPV

∫

∞

∞

ǫi(ω′)

ω′ − ωdω′ (2.1a)

ǫi(ω) = − 1

πPV

∫

∞

∞

ǫr(ω′)

ω′ − ωdω′ (2.1b)

Here, PV stands for the Cauchy principal value of the integral. These relations are very

useful for the determination of frequency dependence of the complex permittivity ǫ(ω).

One can measure the spectral dependence of the absorption, which is related to ǫi and

then ǫr can be calculated using 2.1a and 2.1b.

175

Appendix CInstruments

Some common instruments used in this work are listed below,

• Sputtering system: AJA, ATC orion series uhv, multiple magnetron and RF

sputter coater. For coating ITO for making transparent electrical contacts.

• Optical microscope: Nicon Eclipse digital microscope with a maximum magni-

fication of 100×.

• Scanning electron microscope: Hitachi S-4800 field emission sem.

• Scanning electron microscope/focused ion beam system: Zeiss field emis-

sion SEM/FIB Gemini Nvision 40 CrossBeam.

• Fiber drawing tower: Explained schematically in section 6.1. For the fabrication

of customized photonic crystal fibers and MSIF.

177

Acknowledgement

Before beginning the acknowledgment, I must confess that my writing ability is not that

great. The help and support I received during the tenure of this work deserves much

more appreciation then mentioned here.

First and foremost, I would like to convey my thanks and gratitude to my mentor and

advisor, Prof. Dr. Philip Russell, for giving me the great opportunity to work under his

supervision. I still cherish the memory of the enthusiasm I felt on receiving the email of

my selection in 2006. For the next three and half years, I always felt honored to be part

of his group. His single-minded approach, motivation and brilliant understanding of the

theoretical and experimental aspects of science have definitely changed my perception of

understanding science, specially Photonics. His enthusiasm for work and endless ideas

for solving problems have always influenced me to try new things in experiments and

never be afraid of failure. I would also like to specifically thank him for his cooperation

during the time in which I had to go back to India.

Although, I had very few conversations with my Co-mentor Prof. Dr. Ulf Peschel, but i

would like to thank him for his university lectures, which provided useful insight in the

area of plasmonics.

I would also like to thank Prof. Nicolas Joly, whose special brilliance in fiber fabrication

was a great help in drawing the fibers. His cheerful attitude was always a stress-buster

and it provided a good, humorous atmosphere. I would also like to thank him for teaching

me some badminton tricks.

I would like to express my cordial and very special thanks to my supervisor Dr. Markus

Schmidt. It is no exaggeration to say that this work would have been impossible without

179

C. Instruments

his constant support, motivation, encouragement, attention, help, advice and patience.

It has been a pleasure to work with him; his eagerness to experiment new ideas was

a constant inspiration. He also helped me a lot in understanding various experimental

observations with his deep knowledge of the field. This work was not a easy ride and at

times it was tough, but he always make me feel relaxed and calm during tough times.

I also would like to apologize for my repetition of mistakes in work (especially writing),

which may have caused him frustration. His help and guidance in writing this draft and

later correcting the drafts of this dissertation deserve special thanks. His happy and

progressive attitude has not only influenced my professional life but has also made me a

better person. I wish him the very best in future and would always be happy to work

with him again.

It was a pleasure to work with my mates in the nanowire group; Luis Prill, Howard Lee,

Nicolai Granzow, Patrick Uebel. I had some excellent time working with them. Firstly

thanks must go to Luis Prill for his help in experiments and his help in designing the filling

methods during my initial days. I would like to thank Howard Lee for spending time

with me on experiments and having useful discussions on simulations. My appreciation

also goes to Nicolai Granzow, not only for useful discussion related to work but also for

his help in solving many problems related to latex and for proofreading the text of this

thesis. We also share a common hobby of cooking and different recipes were discussed

between us. I would like to thank Patrick for his help, especially for his useful tips in

understanding the analytical calculations. Although, Arian stayed in this group for a

short period, his knowledge in understanding the instruments was very helpful.

My special thanks must go to the fiber fabrication team of Dr. Michael Scharrer, Silke

Rammler and Fehim Babic. They were crucial in designing and fabricating the fibers. I

have shared some great moments in this group and all the members are wonderful peo-

ple who have helped in many ways. My best greetings and thanks to each one of them

for their support and help. Special thanks must go to my office-mates who provided a

creative, cheerful and motivating environment; Johannes Nold, Dr. Leyun Zang, Mo-

hiudeen Azhar, Sarah Unterkofler, Thang Nguyen, Nicolai Granzow and Patrick Uebel.

I would especially like to mention Dr. Tijmen Euser, Dr. Alexander Podlipensky, Dr.

Holger, Dr. Gordan Wong, Dr. Xin Jiang, Dr. Myeong Soo Kang, Jocelyn Chen, Amir

Abdolvand, Christine, Marta Ziemienczuk, Anna Butsch, Sebastian Stark and Martin

Garbos for their support during my stay in the group; thanks a lot to all of you and my

best wishes for your successful future.

180

My thanks also goes to Bettina Schwender for organizing the administrative aspect and

helping me for various non-technical issues. I would also like appreciate the infrastructure

staff of the institute for their constant support. I would also like to thank IMPRS for

funding my research; I especially appreciate the extremely supportive IMPRS coordina-

tors Heike Auer and Dr. Carsten. I would also like thank Dr. Sven Burger for his help

and useful tips in carrying out FEM simulations using JCMWave.

I would also like to thanks IMPRS, Max Planck society and university of Erlangen for

funding my PhD at various stages.

This acknowledgment is incomplete without appreciating the gratitude and dedicated

support from my family. This work is specially dedicated to my elder brother, whose

endless support is one of the reasons I was able to complete this work. I do not have

words to express my gratitude and thanks to my father and mother, who have been a

pillar of strength and a source of inspiration to me over the years. I would also like to

share my appreciation for my sister-in-law for her support. How can I forget to mention

my sweet baby doll “Anusha” ?, whose one smile is enough to forget the hardships of

work. Lots of love to you, Anusha. One name which I would especially like to mention

here is of my most consistent friend of life Dr. Mani Agrawal. She has always supported

and motivated me to move ahead in both my professional life and and my personal life.

I would also like to thank my friend and sister Reeti Singh for her help during my stay

in Germany. I would also like to use this opportunity to thank all my friends for making

this life a wonderful experience, especially Amit chawla, Nitin Jain, Mohiudeen Azharand

Amit Garg. I would also like to express my thanks to Jennifer for her help in proofreading

this work.

In the end, I would like to thank almighty God for giving me strength to accomplish this

special part of my life.

181

List of Figures

2.1. Real and imaginary part of the complex permittivity around a resonance

frequency ω0 calculated with the Lorentz oscillator model. The black

dashed line corresponds to resonance frequency ω0. The green and blue

dashed line correspond to the higher and lower frequency limit of Re(ǫ). . 15

2.2. Schematic diagram showing a surface with current density K and charge

density ρ divides the two mediums. n is the unit vector normal to the

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3. Schematic design of planar waveguide. A layer of material ’2’ is sandwiched

between two layers of material ’1’. . . . . . . . . . . . . . . . . . . . . . . 23

2.4. Normalized propagation constant β/k0 versus V for TE0, TM0, TE1 and

TM1 modes for planar dielectric waveguide. The green dot corresponds to

V = π/2 [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5. Dispersion (red curve) of a planar SPP mode for a lossless metal. The λp(140nm for silver) is the plasma frequency. The black curve indicates the

dispersion of silica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6. Schematic design of the cylindrical step index waveguide. The material

D with permittivity ǫD is the cladding and material M is the core of the

waveguide with permittivity ǫM . The light is propagating along the z axis

and a is the radius of the core. . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7. Dispersion relation for some lowest order modes (HE11, TE01, TM01, EH11

and HE12) for a step index waveguide with dielectric core. . . . . . . . . 35

2.8. Dispersion relation for different modes of a silver wire (diameter ≈ 1µm)

embedded in silica. The black line is the dispersion of silica and numbers

indicate the SPP mode orders. . . . . . . . . . . . . . . . . . . . . . . . . 36

183

List of Figures

2.9. Schematic of a spiraling SPP which follows a helical path around a Au-wire

embedded in silica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10. Comparison of the exact calculations (solid lines) with the spiraling plas-

mon model (dashed lines) for the dispersion of a silver wire. The numbers

on each line indicate the SPP mode orders. . . . . . . . . . . . . . . . . . 38

2.11. Schematic design of the modified step index fiber (MSIF). . . . . . . . . 41

2.12. Refractive index variation of silica and 16 mol.% GeO2-doped silica with

wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1. Diamond crystal structure with lattice spacing a. . . . . . . . . . . . . . 44

3.2. First Brillouin zone (Wigner-Seitz cell) of diamond crystal lattice. Impor-

tant symmetry points and lines are also indicated. Points of symmetry

inside the zone is in Greek letters and it is in terms of Roman letters for

the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3. Extended zone representation for E − k relation of Kronig-Penney model

(nearly free electron model). Black dotted curve shows the E-k relation

of a free electron. Solid curve shows the modifications of the parabolic

E(k) dependence for the free electrons at the band edges corresponding

to k = π/a First three Brillouin zones are also indicated.(FB: Forbidden

band, AB: Allowed band) . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4. E − k band diagram characterizing conduction and valance band for ger-

manium and silicon in reduced zone scheme. (a) Germanium (b) Silicon

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5. Schematic diagram representing direct bandgap and indirect bandgap semi-

conductors. Different curvatures in valance band edges can led to light hole

(LH) and heavy hole(HH) bands. Figure also illustrates the concept of hole

as a missing electron from valance band. The wavevector of hole is that

of a missing electron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6. Schematic representation of band diagram, density of states g(E), Fermi-

Dirac distribution f(E) and carrier concentration (n for electrons and p

for holes) for intrinsic semiconductors. . . . . . . . . . . . . . . . . . . . 54

3.7. Dependence of intrinsic carrier concentration ni on temperature T for Sil-

icon (Si) and Germanium(Ge). . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8. (a)Variation of lattice constant a as function of temperature for germa-

nium. (b) Temperature dependence of thermal expansion coefficient αth

for germanium [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

184

List of Figures

3.9. Temperature dependence of minimum energy band gap (Eg) for silicon and

germanium. Table in the inset of the plot shows the values of coefficients

for silicon and germanium used to evaluate the curves from equation 3.31. 60

3.10. Temperature dependence of majority carrier concentration in a doped

semiconductor.(n-type silicon with donor concentration Nd = 1015/cm3) . 61

3.11. Intrinsic conductivity σ of germanium as a function of reciprocal of tem-

perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.12. Spectral dependence of optical absorption of germanium for a range from

1eV to 10eV as reported by Philip and Taft at 300K for single crystal

material [48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.13. (a) Absorption spectra for the single crystal germanium for low energy

range of 0.5eV to 2eV as reported by Newman and Dash (1955) at 300K

and 77K [49]. (b) Expanded view of the left curve. . . . . . . . . . . . . 64

3.14. Optical reflectance spectra for germanium single crystal at 300K [51] . . 69

3.15. Schematic diagram showing various radiative combination processes. (1)

Intraband transition within the conduction band, (2)Interband transition,

(3) Decay of an exciton which leads to the emission of a photon, (4)

Donor to free-hole recombination, (5) Free electron to acceptor transi-

tion, (6) Donor to acceptor pair transition (7) Excitation and radiative

de-excitation of electrons from an impurity state which has completely

filled inner shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.16. Schematic diagram showing the process of photoconductivity . . . . . . . 76

4.1. Schematic representation of the curvature of two different types of liquids

inside the glass capillary. (a) Concave curvature (b) Convex curvature. . 80

4.2. Variation of filling length with applied pressure for different diameters for

germanium-melt. These curves are obtained by using equation 4.6. . . . . 84

5.1. (a) Transmission spectrum for a 1 cm thick Suprasil Heraeus sample. (b)

Attenuation coefficient spectra for fused silica provided from Heraeus. . 85

5.2. Variation of refractive index with wavelength for fused silica using equation

5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3. (a) Dielectric function (real and imaginary) of germanium as a function

of wavelength [48]. (b) Refractive index of germanium as a function of

wavelength (Dispersion curve of germanium) [48]. . . . . . . . . . . . . . 89

185

List of Figures

5.4. Dielectric function (real and imaginary) of gold(Au) as a function of wave-

length [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1. Schematic diagram showing various steps for stacking procedure in PCF

fabrication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2. (a)Cane drawing tower. (b) Fiber drawing tower. . . . . . . . . . . . . . 98

6.3. Schematic diagram showing various steps for fiber drawing procedures. . 100

6.4. (a) Solid core endlessly single mode PCF. (b) Hollow core PCF. (c) Pola-

rization maintaining (big hole) PCF. (d) Polarization maintaining (small

hole) PCF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5. Schematic diagram showing various steps of drawing a modified step index

fiber (MSIF). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.6. Schematic diagram showing three basic mechanisms of fiber post processing.105

6.7. Schematic diagram of fiber tapering rig. . . . . . . . . . . . . . . . . . . 106

6.8. Schematic diagram of a filament based splicer. . . . . . . . . . . . . . . . 108

6.9. Schematic diagram showing various steps involving selective hole opening

procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.10. Optical microscope/SEM pictures of end face of ESM PCF at various

stages of selective hole inflation. (a) Original ESM PCF.(b) With a layer

of glue (step 1).(c) After focusing laser light on the a selected hole (step

2) (b) After post-processing the end face with one open hole (step 6). . . 111

6.11. Schematic diagram of Ar purged high temperature pressure cell used to

fill germanium in hollow channels of PCF/MSIF. . . . . . . . . . . . . . 113

6.12. SEM image of some germanium-filled ESM structures using pressure cell

technique. (a) completely filled (first ring). (b) Single hole filled ESM PCF.115

6.13. Schematic diagram showing various step for filling the hole of MSIF with

germanium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.14. SEM image of Ge-filled fibers fabricated using splice filling technique. (a)

PM PCF (b) MSIF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.15. Schematic diagram showing various steps of direct drawing based filling

method. The diagram shows the steps for the fabrication of Au- filled MSIF.119

6.16. Schematic diagram showing the filling of a cane with gold using vacuum. 120

6.17. SEM image of gold-filled capillary with only one hole fabricated using the

direct drawing method. For SEM image of Au-filled MSIF please check

figure 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

186

List of Figures

6.18. Schematic diagram of the experimental set up used to measure the con-

ductivity of germanium/gold -filled PCF, MSIF or a single hole capillary. 122

6.19. Schematic diagram of the experimental set up used to investigate the trans-

mission spectrum of germanium-filled samples. . . . . . . . . . . . . . . . 124

6.20. Schematic diagram of the experimental setup for the measurement of pho-

tocurrent from a Ge-filled MSIF using side pumping with verdi laser. The

operating wavelength of the Verdi laser is at 532nm. . . . . . . . . . . . . 126

6.21. Schematic diagram of the experimental setup for the measurement of pho-

toconductivity from a Ge-filled MSIF using a multi-color He-Ne laser. The

operating wavelength used for the experiment is 543nm. . . . . . . . . . . 128

7.1. Micro-Raman spectrum (a) silicon wire of diameter 90µm (b) germanium

wire of diameter 1.6µm, measured through the side of the Ge/Si filled

capillary. The value of reported Raman peak for bulk single crystalline

Si/Ge is also mentioned in respective plots. . . . . . . . . . . . . . . . . . 133

7.2. (a) SEM image of first ESM fiber (pitch: 2.9 µm, hole diameter: 1.0 µm)

filled with germanium. (b) Normalized transmission spectrum of the sample.134

7.3. (a) SEM image of ESM PCF with a single hole adjacent to the fiber core

filled with germanium. (b) Enlarged view of the glass-core and single

wire of germanium-filled in ESM PCF. The axis indicates the direction of

polarization of the incoupled light. . . . . . . . . . . . . . . . . . . . . . 135

7.4. Experimental (a) and simulated (b) transmission spectra for x- and y-

polarization (Tx and Ty respectively) for structure shown in figure 7.3.

The inset of the left figure shows the CCD image of the transmitted mode

pattern for y-polarization. For simulations, the fundamental mode of the

glass core was selected for numerical investigations. The inset of the right

figure shows the selected mode (axial component of the Poynting vector

for y-polarization) for simulations at a wavelength of 550nm. . . . . . . . 136

7.5. (a) Transmission as a function of wavelength for various positions of the

output polarizer. (b) Ratio (in dB) between the minimum and maximum

transmission as a function of wavelength. The inset of right figure shows

the transmission versus angle of the output polarizer at a fixed wavelength

≈ 950nm. These measurements are obtained for the structure shown in

figure 7.3 with 1.7mm length of Ge-wire. . . . . . . . . . . . . . . . . . . 137

187

List of Figures

7.6. Loss spectra of the structure shown in figure 7.3 for Ge-wire of the length

of 0.8mm. (a) Experimental (black curve) and simulated (right curve)

spectra of the glass-core mode of the structure for x-polarization. (b)

Experimental (black curve) and simulated (right curve) spectra of the

glass-core mode of the structure for y-polarization. The labels on the

peaks of refer to the resonances on the Ge-wire that phase match to the

glass-core mode. Three encircled points correspond to the modes whose

Poynting vector distribution is shown in figure 7.7. . . . . . . . . . . . . 138

7.7. Axial Poynting vector distributions in the vicinity of Ge-wire (top row

form a-c) in a filled ESM PCF calculated by using finite-element simula-

tions. The bottom row (from d-f) shows the axial Poynting vector for a Ge

wire embedded in silica capillary, calculated by directly solving Maxwell’s

equations for the Mie resonances on the wire. The wire radius in both

cases is R = 0.85µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.8. Simulated loss (in dB) spectra for fundamental and first higher order mode

supported by the glass-core in x- and y- polarizations of the structure

shown in figure 7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.9. Variation of resonance wavelengths (loss peaks shown in figure 7.6(b)) with

respect to wire diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.10. (a) The loss spectra (dB/cm) of the fundamental TE and TM guided

modes in silica layer (4µm wide) and a layer of Ge (1µm wide), sandwiched

between cladding material of refractive index 1.35. (b) Modulus square of

the electric fields for four modes marked as A, B, C and D in left figure. . 143

7.11. Effect of imaginary part of dielectric function on the loss spectra for struc-

ture shown in figure 7.3. The curves are plotted for ǫi (blue) ǫi/2 (red)

and ǫi/3 (black). The real part of dielectric function was unchanged in

these calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.12. Ovelap integral of transverse components in y-polarization at some specific

wavelengths. (a) Magnetic field components Hx and Hy (b) Electric field

components Ex and Ey. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.1. SEM image of gold-filled MSIF polished using focused ion-beam milling.

The wire diameter is ≈ 3.8µm. The coordinate system defines the direction

of the electric field in two principal states of polarization. . . . . . . . . . 148

188

List of Figures

8.2. (a) Measured attenuation spectra of light guided in the glass core of the

gold-filled MSIF (red, y-polarization; black, x-polarization). (b) Corre-

sponding finite-element simulation of the loss of the MSIF; parameters are

given in the text (red: y-polarization; black: x-polarization). The spectral

positions of the loss peaks are also mentioned for both the cases. . . . . . 149

8.3. Real part of the effective refractive indices of guided SPP modes on isolated

gold wires embedded in fused silica. The numbers refer to the mode order.

The magenta curve is the dispersion of the fundamental glass-core mode.

(b) and (c) shows the axial Poynting vector distributions of the guided SPP

modes at the crossing points marked as blue dots (908nm and 1010nm). . 151

8.4. SEM image of a small hole PM PCF, in which a single elliptical small hole

(0.85µm and 1.23µm as minor and major axis, respectively) adjacent to

the core is filled with gold. The rectangular core of PM PCF consist has

dimensions of 3.57µm and 1.1µm as long and short side of rectangle. . . 152

8.5. (a) Measured transmission spectrum in y-polarization of a 6mm (blue

curve) and 24.5mm (orange curve) of length of the structure shown in

figure 8.4. The mode number marked on each transmission dip corre-

sponds to the modes of the gold wire to which the glass-core mode is phase

matched. (b) The dispersion curves of pure silica (red curve), unfilled PCF

(black curve) and different guided SPP modes of a single Au-wire embed-

ded in silica. The different Au-wire mode orders are marked with their

corresponding mode numbers, i.e. 2, 3, 4, 5. . . . . . . . . . . . . . . . . 153

8.6. Experimentally observed optical near field images of the fiber end face at

two different wavelengths (A) 546nm and (B) 546nm, superimposed onto

the SEM image of the filled PM PCF. The position of the wavelengths are

also marked in transmission spectra shown in figure 8.5. . . . . . . . . . . 155

8.7. (a) Measured transmission spectra (in dB) for the two polarization stateS

for the structure shown in figure 8.4. (b) The FEM simulation of the

structure. For convenience the transparency (cm/dB) of the glass-core

mode is plotted for simulations. The dashed line in figure (a) corresponds

to the positions of the dip in simulations. . . . . . . . . . . . . . . . . . . 156

9.1. Measured relative spectral shift of the position of three of the resonance

peaks (TE05 (1180nm), EH15 (1300nm) and EH24 (1390nm) peaks in y-

polarization) is plotted versus temperature. The slopes of the fitted linear

curves are shown in units of nm/K. . . . . . . . . . . . . . . . . . . . . . 158

189

List of Figures

9.2. Variation of photocurrent with incident power for a Ge-filled large pitch

PCF with hole diameter ≈ 1µm and pitch ≈ 7µm. The experiments were

performed for three different circuit voltages. . . . . . . . . . . . . . . . . 160

9.3. (a) SEM image of the MISF strcture (core diameter ≈ 1µm and Ge-wire

diameter ≈ 1.3µm) used to measure photoconductivty. (b) Transmission

spectrum (in dBm) in x- and y- polarizations for the structure shown in (a).161

9.4. Change in photocurrent (in nano-amperes (nA)) as a function of absorbed

power (in µW) in the Ge-wire for x- and y- polarizations of the incident

light. The straight lines are the linear fit to the measured data points. . . 163

9.5. Change in resistivity ∆ρ (in Ω-cm) as a function of absorbed power in

the Ge-wire for x- and y- polarization of incident light. The straight lines

shown the linear fit to the measured data points. . . . . . . . . . . . . . 165

9.6. Calculated dispersion coefficient and loss for a step waveguide with pure

germanium-core and silica-cladding (a) Dispersion coefficient (in ps/(nm-

km)) (b) Loss in (dB/mm). . . . . . . . . . . . . . . . . . . . . . . . . . 167

190

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Publications

Journal publications

1. H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St.J.

Russell. Plasmon resonances on gold nanowires directly drawn in a step-index fiber.

Opt. Lett., 35(15):2573–2575, 2010.

2. H. K. Tyagi, M. A. Schmidt, L. Prill Sempere, and P. S. Russell. Optical pro-

perties of photonic crystal fiber with integral micron-sized ge wire. Opt. Express,

16(22):17227–17236, 2008.

3. H. W. Lee, M. A. Schmidt, H. K. Tyagi, L. Prill Sempere, and P. St. J. Rus-

sell. Polarization-dependent coupling to plasmon modes on submicron gold wire in

photonic crystal fiber. Applied Physics Letters, 93(11):111102, 2008.

4. M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J.

Russell. Waveguiding and plasmon resonances in two-dimensional photonic lattices

of gold and silver nanowires. Phys. Rev. B, 77(3):033417, Jan 2008.

199

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